research - based multiple - choice tests can be powerful probes of student understanding .they are easy and economical to administer and to grade , have objective scoring , and are amenable to statistical analysis that can be used to compare student populations or instructional methods .a major drawback is that the thought processes are not revealed completely by the answers alone .however , when combined with student interviews , well - designed tests are powerful tools for educational assessment .conceptual multiple - choice tests have already been designed to assess student understanding of force , kinematics graphs , heat , circuits , and electricity and magnetism. they suggest that students knowledge of physics often is fragmented and context - dependent , and that students have many common difficulties .energy and momentum are two of the most fundamental concepts in physics. in the lower - level physics courses , they are typically introduced after newton s laws .we have designed and administered a research - based 25-item multiple - choice test and conducted individual interviews to explore student understanding of energy and momentum concepts. we are interested in understanding the difficulties students have in interpreting these concepts and in correctly identifying and applying them in different physical situations .we also wish to know the extent to which the difficulties are universal , and if there is a correlation with student preparation ( for example , calculus or algebra background ) .the identification of student difficulties can help in designing instructional tools that address them .part of the rationale for combining energy and momentum concepts is to investigate the extent to which students can identify the relevant concept in a particular situation .during the test design , we paid particular attention to the important issues of reliability and validity. reliability refers to the relative degree of consistency in scores between testing if the test procedures are repeated for an individual or group .validity refers to the appropriateness of the test score interpretation .the test design began with the development of a test blueprint that provided a framework for planning decisions about the desired test attributes .we tabulated the scope and extent of the content covered and the level of cognitive complexity desired .the specificity of the test blueprint was useful for creating questions .the energy concepts included the work - energy theorem , conservation of mechanical energy , and work done by gravitational and frictional forces .the momentum concepts included the definition of momentum , impulse - momentum theorem ( impulse was defined explicitly in the test ) , and conservation of momentum with examples from elastic and inelastic collisions .we also planned to evaluate student understanding of the concept of system " in various contexts .we also planned to include one question that explicitly required the application of both energy and momentum concepts .energy and momentum questions pertaining to simple harmonic motion , explicit mention of conservative and non - conservative forces and center - of - mass reference frame were deliberately excluded , because we wanted the test content to be commensurate with the curriculum in most calculus- and algebra - based courses for science and engineering majors .we classified the cognitive complexity using a simplified version of bloom s taxonomy: specification of knowledge , interpretation of knowledge and drawing inferences , and applying knowledge to different situations .then , we outlined a description of conditions / contexts within which the various concepts would be tested and a criteria for good performance in each case .the tables of content and cognitive complexity along with the criteria for good performance were shown to five physics faculty members at the university of pittsburgh for review .modifications were made to the weights assigned to various concepts and to the performance criteria based upon the feedback from the faculty about their appropriateness .the performance criteria was used to convert the description of conditions / contexts within which the concepts would be tested to make approximately 50 free - response questions .these questions required students to provide their reasoning for their responses .the free - response questions were administered ( in groups of 10 or 20 ) to students in the calculus- and algebra - based courses at the university of pittsburgh in the fall of 1998 . often , some students in a class were given one set of questions and others were given another set in order to sample student responses on most of the questions .we also tape - recorded interviews with 10 introductory student volunteers using the think - aloud protocol. forty multiple - choice items ( questions ) were then designed using the most frequent student responses for the free - response questions and interviews as a guide for making the alternative distractor choices .four alternative choices have typically been found to be optimal  and we chose the four distractors to conform to the common difficulties to increase the discriminating properties of the items .ten physics faculty members and postdocs were asked to review the multiple - choice items and comment on their appropriateness and relevance for introductory physics and to detect ambiguity in item wording .an item review form was developed to aid the faculty in reviewing the items .the faculty also classified each item on a scale from very appropriate to least appropriate .further modifications were made based upon their recommendation .then , a multiple - choice test was assembled using 28 items which closely matched the initial table delineating the scope of the content and cognitive complexity .the same faculty members who reviewed the items were shown the test and some modifications were made .the test was administered as a 50-minute post - test ( after instruction ) to several hundred students in calculus- and algebra - based courses at the university of pittsburgh in spring 1999 .seven student volunteers who had taken the test were interviewed using the think - aloud protocol .item analysis of student responses was performed to judge the quality of each item .in addition to the calculation of item difficulty and discrimination, the item analysis included creating a table to count the number of students selecting each distractor in the upper and lower quartiles .this analysis was very useful to determine whether individual items and distractors functioned as expected .based upon the item analysis and interviews , the test items were modified further .the number of items in the test was decreased to 25 .including the spring and fall 1999 , 2000 and 2001 the test has been administered to more than 3000 students from approximately 30 calculus- and algebra - based courses in different colleges and universities .some classes administered the test both as a pre-/post - test to assess students conceptions of energy and momentum before instruction , the effectiveness of the instruction , and the effect of pre - test on post - test .every semester an item analysis was performed after its administration , a few students were interviewed , and some items were slightly modified .we have conducted a total of 34 one - hour interviews during the test development process . in fall 2000 , the test was administered to graduate students enrolled in a teaching methods class .their average score was greater than 80% and the reliability coefficient ( which is a measure of the internal consistency of the test ) was greater than 0.8 .the final version of the test is included in appendix [ appa ] .the test was administered before the instruction on energy and momentum concepts ( pre - test ) only in some classes .the average pre - test scores typically ranged from 24%36% depending upon the class type , for example , calculus versus algebra ( higher for calculus ) , and whether the test was administered before or after any instruction on linear kinematics / dynamics ( some questions can be answered without explicitly invoking energy or momentum concepts ) .we first discuss the post - test results of the version administered to 1356 students in fall 2000 .the reliability coefficient for the calculus - based classes was slightly more than 0.75 ( 1170 students ) , while that for the algebra - based classes was 0.68 ( 186 students ) .figures 1 and 2 show the post - test statistics for the calculus - based courses ( 1170 students ) including the honors ( two ) and active - engagement classes ( one ) .the average post - test score was 49% ( standard deviation 18% ) .figure 1 shows a plot of the item difficulty ( the fraction of students who got the item correct ) , which ranged from 0.2 to 0.8 .this range is consistent with our goal because we want to explore student performance on items with a variety of cognitive complexity .figure 2 shows a plot of the point biserial discrimination ( pbd) for individual items which measures the ability of an item to distinguish between high ( those who performed well on the test overall ) and low performing students .the pbd ranged from 0.21 to 0.48 with only one item with pbd less than 0.28 , which is good by the standards of achievement test design. [ xx it looks like you left out some references to table i and ii which were in the hard copy from redish xx ] the average score on the items focusing mainly on the energy concepts ( 14 total ) was 46% ( 20% ) , while that on items focusing mainly on momentum concepts ( 10 items ) was 55% ( 22% ) .there was a reasonable correlation between the performance on the energy questions and momentum questions ( pearson correlation 0.54 ) .we performed a factor analysis ( principal component method) which identified six factors with a variance greater than 4% .however , the largest of these factors accounted for less than 15% of the variance .this variance is very small for the first factor so none of the factors are meaningful .although the analysis of variance ( anova) shows statistically significant differences between several calculus - based classes at the level of , the effect size is small ( ) for all pairs except those involving an honors class or an active - engagement class. also , anova shows a statistically significant difference between the calculus - based ( excluding the honors and active - engagement classes ) and algebra - based classes in terms of the total average score , but again the effect size is small .anova on individual items shows that the differences between the calculus- and algebra - based classes are not statistically significant for 12 of the 25 test items .   to assess the effect of instruction , we now discuss results ( unmatched ) from three calculus - based classes for which both pre-/post - tests were administered .a total of 352 and 336 students from the three classes took the pre - test and post - test in fall 2001 and the average scores were 33.6% and 51.7% respectively .the average post - test score was more than 50% on 15 out of the 25 items .table i shows the average percent correct on the pre-/post - test and the normalized gain , where and are the pre-/post - test scores in percent) on each test item separately for students whose total test scores were in the upper 25% ( group h ) , lower 25% ( group l ) , and all students ( a ) .the average pre - test scores of students in the upper and lower quartiles were 49.8% and 19.2% and the average post - test scores in these two groups were 70.6% and 32.2% respectively .thus , students in the upper quartile gained significantly more than those in the lower quartile with an average gain of 0.45 for group h , 0.16 for group l , and 0.29 for all students a. the upper and lower quartiles were selected based the average scores on the pre and post tests .table ii shows the distribution of alternative choices for the pre-/post - tests .it shows the types of preconceptions students had before instruction on the energy and momentum concepts and how the conceptions changed after the instruction .comparison of the pre-/post - tests in table ii shows that instruction improved student conceptions in some cases , but not in others .the test results reveal that students lack a coherent understanding of energy and momentum concepts and have difficulty applying them to different physical situations .they often focused on the surface features and were distracted by irrelevant details. during the interview , some students suggested that the lack of numerical values for the parameters prevented them from checking if they are on the right track .many students did not realize that work and energy are scalar quantities and momentum is a vector quantity .questions involving work - energy or impulse - momentum theorems were typically perceived to be more difficult than those involving their special cases : the conservation of mechanical energy and momentum respectively .however , students had great difficulty in using the conservation principles appropriately in many situations . in the following ,we discuss some insightful features of student responses by loosely clustering items based upon the relevant physics concepts . although the same item can be categorized in different ways , we choose to discuss them within a particular cluster .  .items 2 , 4 , 15 , and 22 can all be answered based upon the conservation of mechanical energy , because there is no work done by non - conservative forces .( the normal force is present in the slide " related items ( 2 ) and ( 15 ) , but it is perpendicular to the direction of motion of the object , so it does no work on it . )students performed very differently on these items and did not perform any better on items ( 4 ) and ( 22 ) which involved only the gravitational force compared to items 2 and 15 which had a normal force present .interviews show that in many physical situations students knew what " but did not know `` why . '' for example , in response to item 4 , several interviewed students said that the balls thrown from a cliff will reach the ground at the same speed , regardless of the angle of projection , but they could not justify their answers based either upon energy or kinematic considerations . item 15elicits strong belief that the heavier person has a larger speed at the bottom of the slide because the larger weight causes a greater acceleration , " a belief which did not improve much after instruction .an interviewed student noted : _i remember my dad used to slide down faster because his mass carried him down more_. another student who chose ( c ) used correct reasoning except in the last step where he did not realize that the mass will not affect the speed : _ another way to see it is that your potential energy is larger than your niece s your kinetic energy will be larger at the bottom so your speed is larger_. if it were a numerical problem , there is a chance that this student would have solved it correctly , perhaps without consciously noting that the mass drops out . for item 22 ,interviews indicate that students who viewed this as a projectile motion problem were very likely to get it wrong . a student who chose ( a ) explained : _ ball ( 1 ) gets there fastest so it will not lose its speed as much and for the same reason ball ( 2 ) will be faster than ball ( 3)_. another student had something similar to say : _ the longer the path more chance gravity has to slow the ball down_. a student who chose ( c ) explained : _ ball thrown vertically will slow down most because all its motion is in the y direction the more is the component in the x direction the more it remains unchanged_. the student is probably confusing the time it takes to reach a height with the speed at that height , and he is overlooking the fact that all the three balls are at the same height . another student who chose ( c ) explained : _ smaller angle _ [ with horizontal ] _ will reduce downward acceleration making the speed larger_. incidentally , in item 22 , in the post - test , only 10% from groupl chose ( e ) indicating that the speed of a ball at a given height depends on the mass , but 60% indicated in item 15 that the speed of an object sliding down the frictionless slide depends upon mass .this context dependent response suggests that students knowledge is fragmented and their recall is triggered by surface features rather than the underlying physics concepts .items 1 , 6 , 8 , and 20 probe understanding of work done by the gravitational force .response to item 1 shows the pervasive belief that the work done by the gravitational force is path - dependent .part of the problem is the difference between the physical and everyday definitions of work . in everyday language ,more work is considered to be done if the person becomes more tired afterward .item 6 can most easily be answered by invoking the definition of work as the scalar product of force and displacement and realizing that because the angle between the force and displacement is , the work must be zero .it can also be answered by using the work - kinetic energy theorem .student response to item 6 shows that instruction was more successful in helping group h students understand the difference between velocity and speed than group l students .item 8 assesses students ability to differentiate between an applied force and the work done by it .many incorrectly reasoned that because a smaller magnitude tension force is required along an inclined plane , the magnitude of the work done by tension or the gravitational force must therefore be smaller .the ability to distinguish between the force and the work done by the force did not improve at all post instruction for group l students . as mentioned earlier, part of the problem may be students inability to grasp the distinction between the term work as used in physical and in everyday language .an interviewed student who chose ( d ) noted : _ you do a lot less work in the incline case than in the vertical case it is called mechanical advantage or something _ when asked how he would calculate the work he added : _ i do nt know if i remember the math the concept is clear to me though you definitely do less work when it is inclined_. another student had a similar explanation : _ in inclined case it would be easier to pull which implies less work for you_. a third student explained : _ tension in case ( ii ) is at an angle say so the work done will have and it will be smaller than straight up_. incidentally , the number of students from group l who believed that the work done by the gravitational force is path dependent in item 1 is significantly larger than in item 8 and this difference in number suggests that students knowledge is fragmented and context - dependent.  items 9 , 12 , 24 , and 25 probe the understanding of work done by non - conservative forces . in item 9 , most of the improvement for group h from the pre - test to post - test was due to the transition from choice ( b ) to ( a ) , because the students learned that a constant speed going up the hill does not imply that the mechanical energy is conserved .the trend is opposite for group l. a student who chose ( e ) explained : _ if you ignore the retarding effects of friction mechanical energy will be conserved no matter what_. other interviewed students who chose ( e ) also suggested that the retarding effect of friction was the only force that could change the mechanical energy of the system . although some students may have chosen ( b ) because they could not distinguish between the kinetic and mechanical energies , the following interview excerpt shows why that option was chosen by a student despite the knowledge that kinetic and mechanical energies are different : _ if she goes up at constant speed then kinetic energy does not change that means potential energy does not change so the mechanical energy is conserved mechanical energy is kinetic plus potential_. the student continued when asked to explain what the potential energy is : _ uhh is nt it ?_. when asked to explain why it is not changing , the student first paused and then added : _  is the height i guess does change if she goes up the hill hmm may be that means that potential energy changes .i am confused i thought that if the kinetic energy does not change , then potential energy can not change are nt the two supposed to compensate each other is it a realistic situation that she bikes up the hill at constant speed or is it just an ideal case ? _the student is convinced that the mechanical energy is conserved when the bike goes up at a constant speed , and he initially thinks that both the kinetic and potential energies must remain unchanged .when he confronts the fact that the potential energy is changing , instead of reasoning that the mechanical energy must be changing if the kinetic energy is constant , he thinks that it is probably not realistic to bike up the hill at a constant speed .he wonders if it is only possible in the idealized physics world .although he ignores the work done by the non - conservative force applied on the pedal to keep the speed constant , his statements shed light on student s epistemological beliefs about how much one can trust physics to explain everyday phenomena .a student who chose ( c ) provided an interesting explanation : _ in case ( 1 ) the kinetic energy is transferred to potential energy so the mechanical energy is conserved and in case ( 2 ) obviously if the speed is constant mechanical energy is conserved .case ( 3 ) is out because she is accelerating ._ this example shows student s inconsistent reasoning and use of the term acceleration . actually , there is acceleration not only in case ( 3 ) but also in case ( 1 ) ( slowing down ) , but the student ignores it in case ( 1 ) . at the same time , he put cases ( 2 ) and ( 3 ) in different categories although the cyclist was pedaling in both cases . for item 12 ,the strongest distractor was the belief that there is a net force in the direction of motion. this belief was essentially unchanged by instruction for groupl while half of group h benefited from instruction . items 24 and 25 can both be answered using the work - kinetic energy theorem .item 24 has previously been investigated in depth, and students had great difficulty in realizing that because identical constant forces are applied over the same distance to both masses ( which start from rest ) , their kinetic energies are identical regardless of their masses .some students correctly stated that the velocity of block a will be greater , but they had difficulty in reasoning beyond this .interviews show that the choice ( b ) was often dictated by the fact that the kinetic energy increases as the square of the speed but only linearly with mass. in item 25 , all distractors are dimensionally _ incorrect _ and equally popular .items 13 and 17 probe understanding of the conservation of mechanical energy and related issues . in item 13 ,some alternative choices used energy concepts while others used momentum concepts to assess how well students recognize the relevant concepts . in an interview , a student explained : _( c ) and ( e ) are obviously not right because they are the same momentum and kinetic energy both depend on mass and speed i think it is ( b ) because this one falls from a greater height_. the student has forgotten about the vector character of momentum .interviews show that at least some students who chose ( a ) were confident that the kinetic energy has a direction ( is a vector ) and the kinetic energies of the two carts cancel each other . in item 17, students who chose ( d ) believed that the work done by the gravitational force on the ball falling from the tower is negative .interviews show that many students did not invoke physics principles to come to this conclusion ( for example , the basic definition of work ) , but thought that the work must be negative if the ball is falling in the `` negative y direction .'' students who focused on speed rather than kinetic energy were likely to get confused between options ( a ) and ( b ) .an interviewed student who chose ( a ) started with the following correct statement : _ is nt it true that the velocity of the ball increases by like 9.8 m/s every second kinetic energy is _ ( writes down the formula ) .but later he got misled due to a faulty proportional reasoning and added referring to the formula : _ increases by equal amount over equal times so increases by equal amount over equal times mass is not changing _  .item 5 probes the understanding of momentum conservation in an elastic and inelastic collision . in item 5, students choosing the popular distractor ( b ) believed that the block in which the bullet gets embedded moves faster because the bullet transfers all of its kinetic energy to the block in the inelastic collision .the increased frequency of distractor ( a ) from pre - test to post - test shows that instruction was not effective in helping students learn that momentum is a vector quantity and when the bullet s direction changes during the elastic collision , it transfers momentum to the block .another popular distractor ( c ) , which has several buzz - words such as newton s second law " and `` effective mass , '' is meaningless but was designed based upon student interviews .items 3 and 11 were designed partly to evaluate whether students can identify the appropriate system for which momentum ( and also the kinetic energy for elastic collisions ) is conserved and indicated that many students were confused about it .many believed that the momentum and kinetic energy are conserved for each object .incidentally , in the pre - test response to item 11 , 48% noted that a person standing on ice will remain stationary when hit by a horizontally moving ball ( regardless of whether it bounced elastically or not ) , because those choices had several buzz - words .items 10 and 21 probe understanding of momentum conservation in an inelastic collision or explosion . in item 10, many students believed that the kinetic energy is conserved in an explosion .interviews show that some students believed that the kinetic energy is a vector . a student who chose ( a ) noted : _ it does not lose kinetic energy because there is no friction_. item 14 is about the comparison of two completely inelastic collisions , but the alternative choices included both the energy and momentum concepts and students had to determine which was appropriate .even in item 21 , different alternative choices included both momentum and mechanical energy conservation to assess if students could identify the relevant concepts .an interviewed student noted : _i know this is an inelastic collision i will pick ( a ) because rain is falling vertically and is not going to influence the speed of the cart _ . when asked about his thoughts on choice ( c ) , the student paused and then added : _ probably the horizontal momentum is conserved but the rain did not have any horizontal momentum so the velocity of the cart in the horizontal direction is unchanged that s why i picked ( a ) _ the student s last statement is inconsistent because if the horizontal momentum is conserved , then the addition of mass should slow the cart .it is unclear however what the student meant by the horizontal momentum is conserved . "taking item 3 as a guide , the student may very well be just referring to the cart as the system for which the horizontal momentum is conserved , although choice ( c ) explicitly noted the cart - rain " system .items 19 and 23 are applications of the impulse - momentum theorem . in item 19 , students had to realize that the change in momentum is the same whether the biker slams into a haystack or a concrete wall , but the haystack changes the momentum over a longer time and reduces the average force . in the interview , a student who chose ( a ) did not use the appropriate physics principle as the starting point and explained : _ haystack gives you a smaller force so the impulse is smaller_. a student who chose ( e ) noted : _ harder objects give up more energy than softer objects_. in item 23 , the change in momentum of the rubber ball in the elastic collision with the surface is twice compared to that of the putty ball in the completely inelastic collision . because the time over which the velocity of the balls changes from its initial to its final value due to the contact with the surface is the same , the average force on the surface by the rubber ball is twice as much as the putty ball .interviews show that many students who chose distractor ( a ) reasoned that because the masses are the same , the force is proportional to the speed . because both balls have the same speed when they hit the surface , the average force exerted on the surface is the same , regardless of whether the ball bounces back or gets stuck .some students focused on the weights of the two balls and said that the balls would exert the same force on the surface because their weights were the same .some said they did not know why was provided , because the weight does not depend upon time .after instruction , students overall performance improved on all test items ( see gains in table i ) except item 16 on the ballistic pendulum , which was the only one that involved both the energy and momentum concepts simultaneously .interviews show that some students who verbalized that the problem had several parts often ended up focusing on only one aspect .one student who chose ( b ) explained : _ because they are sticking it is a collision so momentum is conserved . _ poor performance on this item , which involved both the energy and momentum concepts ,suggests that even group h had difficulty analyzing multi - part problems involving more than one concept . in the post - test ,student performance on items 7 and 18 , which focused more on the definition and factual information ( lower cognitive complexity ) , was above average . responding to item 7 , an interviewed student gave the following reasoning for why the motorcycle has a larger momentum : _ it has an acceleration which contributes to force which is involved in momentum_.we have developed a research - based test to assess introductory students conceptual understanding of energy and momentum .we found that students have difficulty in qualitatively interpreting the basic principles related to energy and momentum and in applying them in physical situations .the difficulties were not strongly dependent on student population or calculus background , except for honors and active - engagement classes .comparison of the pre-/post - test scores of students in the lower ( l ) and upper ( h ) quartiles based upon the overall test score shows that the upper - quartile group had significantly higher normalized gains .the test can be administered as a pre-/post - test to assess the difficulties prior to instruction , as well as those that remain uncorrected following a particular type of instructional intervention .the test can be used to compare the understanding of energy and momentum concepts in courses employing different instructional designs and strategies .we are very grateful to all the faculty who reviewed the various components of the test at several stages and provided invaluable feedback .we are also very thankful to all the faculty who administered the test .this work is supported in part by the national science foundation and the spencer foundation .* instructions *   1 .you lift a suitcase from the floor to a table .in addition to the weight of the suitcase , select all of the following factors that determine the work done by the gravitational force on the suitcase. 1 .  whether you lift it directly up to the table or along a longer path 2 .  whether you lift it quickly or slowly 3 .the height of the table above the floor 1 .\(1 ) only 2 .\(3 ) only 3 .\(1 ) and ( 3 ) only 4 .\(2 ) and ( 3 ) only 5 .( 1 ) , ( 2 ) and ( 3 ) 2 .two frictionless slides are shaped differently but start at the same height and end at the same level as shown below .you and your friend , who has the same weight as you , slide down from the top on different slides starting from rest .which one of the following statements best describes who has a larger speed at the bottom of the slide ? 1 .you , because you initially encounter a steeper slope so that there is more opportunity for accelerating .you , because you travel a longer distance so that there is more opportunity for accelerating .your friend , because her slide has a constant slope so that she has more opportunity for accelerating .your friend , because she travels a shorter distance so that she can conserve her kinetic energy better .5 .  both of you have the same speed .a moving white hockey puck collides with a stationary red hockey puck on a frictionless horizontal surface .no net external force acts on the two - puck system .select all of the following statements that must be for this elastic collision .the kinetic energy of the white puck is conserved ( same before and after the collision ) .2 .  the linear momentum of the white puck is conserved .the linear momentum of the two - puck system is conserved . 1 .\(1 ) only 2 .\(3 ) only 3 .\(1 ) and ( 2 ) only 4 .\(1 ) and ( 3 ) only 5 .( 1 ) , ( 2 ) and ( 3 ) 4 .  two identical stones , a and b , are shot from a cliff from the and with .stone a is shot vertically up , and stone b is shot vertically down ( see figure ) . which one of the following statements best describes which stone has a larger speed right before it hits the ground 1 .both stones have the same speed .2 .  a , because it travels a longer path .a , because it takes a longer time .a , because it travels a longer path and takes a longer time .b , because no work is done against the gravitational force .two bullets are fired horizontally with at two blocks of .the blocks rest on a frictionless horizontal surface and are made of hard steel and soft wood respectively ( see figure ) .one bullet bounces elastically off the steel block .the other bullet becomes embedded inside the wood block .which one of the following statements best describes which block travels faster after the collision ?1 .  the wood block , because it has gained the momentum of the bullet , while the other bullet does not impart its momentum to the steel block .2 .  the wood block , because the bullet transfers all of its kinetic energy to it .3 .  the wood block , because its larger effective mass after the collision , in accordance with newton s second law , results in a larger force to accelerate the block .4 .  the steel block ,because the bullet bounces off from it .both blocks travel with the same speed .a satellite is moving around earth in a orbit at a ( see figure ) . the only force that acts on the satelliteis earth s gravitational force which points directly toward earth s center .which one of the following statements is as the satellite moves in the orbit ? 1 .  the gravitational potential energy of the satellite decreases as it moves from a to b. 2 .the work done on the satellite by the gravitational force is negative for the motion from a to b. 3 .the work done on the satellite by the gravitational force is zero for the motion from a to b. 4 .the velocity of the satellite remains unchanged as it moves from a to b. 5 .none of the above . 7 .a motorcycle and a truck are both moving in the same direction in adjacent lanes on a highway . at a particular instant , the speed of the motorcycle is four times as large as the speed of the truck .at that instant , the motorcycle is accelerating forward while the truck is moving at a constant velocity .which one of the following statements best describes which vehicle has a at that instant ? 1 .the truck , because it has a larger mass .2 .  the motorcycle , because it is moving faster .3 .  the motorcycle , because it has an acceleration .4 .  the motorcycle , because it is moving faster and also has an acceleration . 5 .  not enough information . 8 .you want to lift a heavy block through a height by attaching a string of negligible mass to it and pulling so that it moves at a .you have the choice of lifting it either by pulling the string vertically upward or along a frictionless inclined plane ( see figure ) .which one of the following statements is ? 1 .the magnitude of the tension force in the string is smaller in case ( i ) than in case ( ii ) .the magnitude of the tension force in the string is the same in both cases .the work done on the block by the tension force is the same in both cases .4 .  the work done on the block by the tension force is smaller in case ( ii ) than in case ( i ) .the work done on the block by the gravitational force is smaller in case ( ii ) than in case ( i ) .three bicycles approach a hill as described below : 1 .cyclist 1 stops pedaling at the bottom of the hill , and her bicycle coasts up the hill .cyclist 2 pedals so that her bicycle goes up the hill at a constant speed .3 .  cyclist 3 pedals harder , so that her bicycle accelerates up the hill .+ ignoring the retarding effects of friction , select all the cases in which the of the cyclist and bicycle is conserved\(1 ) only 2 .\(2 ) only 3 .\(1 ) and ( 2 ) only 4 .  \(2 ) and ( 3 ) only 5 .( 1 ) , ( 2 ) and ( 3 ) 10 . a bomb at rest on a surface explodes and breaks into three pieces that fly apart as shown below .select all of the following statements that must be after the bomb has explodedthe total kinetic energy of the bomb fragments is the same as that of the bomb before explosion .the total momentum of the bomb fragments is the same as that of the bomb before explosion .the total momentum of the bomb fragments is zero . 1 .\(1 ) only 2 .\(2 ) only 3 .\(3 ) only 4 .\(2 ) and ( 3 ) only 5 .( 1 ) , ( 2 ) and ( 3 ) 11 .you and your friend are both standing on a horizontal frictionless surface . to get your friend s attention, you throw a ball due west at your friend as shown below .the ball bounces elastically off your friend s back .which one of the following statements is about this situation ? 1 .your friend will remain stationary because the ball bounces elastically and does not impart its momentum to her .2 .  your friend will remain stationary because the kinetic energy of the ball is conserved in an elastic collision .3 .  your friend will remain stationary due to conservation of both linear momentum and kinetic energy .4 .  you will remain stationary after you throw the ball due to conservation of both linear momentum and kinetic energy .you will move east after you throw the ball due to conservation of linear momentum .12 . using a rope of negligible mass, you pull a box along a horizontal surface with a constant horizontal force .the box moves at a from position a to position b. the force of friction be neglected .which one of the following statements concerning the motion of the box from a to b is ? 1 .the work done on the box by the gravitational force is non - zero .the work done on the box by is positive .the total work done on the box by the net force is non - zero .the magnitude of the work done on the box by is equal to the magnitude of the work done by .the magnitude of is greater than the magnitude of . + * *  + carts a and b are identical in all respects before the collision . + * figure ( i ) * : cart a starts from rest on a hill at a height above the ground .it rolls down and collides head - on " with cart b which is initially at rest on the ground .the two carts stick together . + * figure ( ii ) * : carts a and b are at rest on opposite hills at heights above the ground .they roll down , collide head - on " with each other on the ground and stick together . 13 . which one of the following statements is about the the carts collide in the two cases ? + the collision on the ground , 1 .the kinetic energy of the system is zero in case ( ii ) .2 .  the kinetic energy of the system is greater in case ( i ) than in case ( ii ) .the kinetic energy of the system is the same in both cases .4 .  the momentum of the system is greater in case ( ii ) than in case ( i ) .the momentum of the system is the same in both caseswhich one of the following statements is about the the carts collide in the two cases ? + the collision , 1 .the kinetic energy of the system is greater in case ( ii ) than in case ( i ) .the kinetic energy of the system is the same in both cases .3 .  the momentum of the system is greater in case ( ii ) than in case ( i ) .the momentum of the system is non - zero in case ( i ) while it is zero in case ( ii ) .the momentum of the system is the same in both cases .15 . while in a playground , you and your niece take turns sliding down a frictionless slide .your mass is 75 kg while your little niece s mass is only 25 kg .assume that both of you begin sliding .which one of the following statements best describes who has a larger speed at the bottom of the slide ? 1 .both of you have the same speed at the bottom .2 .  your niece , because she is not pressing down against the slide as strongly so her motion is closer to free fall than yours . 3 .you , because your greater weight causes a greater downward acceleration .4 .  your niece , because lighter objects are easier to accelerate .you , because you take less time to slide down .two small spheres of putty , a and b , of equal mass , hang from the ceiling on massless strings of equal length . sphere a is raised to a height as shown below and released .it collides with sphere b ( which is initially at rest ) ; they stick and swing together to a maximum height .the height can be determined in terms of by invoking which of the following principles : 1 .  the conservation of mechanical energy 2 .the conservation of linear momentum 1 .\(1 ) only 2 .\(2 ) only 3 .  both ( 1 ) and ( 2 ) 4.  either ( 1 ) or ( 2 ) but not both 5 .  principles ( 1 ) and ( 2 ) alone are not sufficient to find in terms of .you drop a ball from a high tower and it falls freely under the influence of the gravitational force .which one of the following statements is ? 1 .  the kinetic energy of the ball increases by equal amounts in equal times .2 .  the kinetic energy of the ball increases by equal amounts over equal distances . 3 .there is zero work done on the ball by the gravitational force as it falls .the work done on the ball by the gravitational force is negative as it falls .the total mechanical energy of the ball decreases as it falls .+ * in questions ( 18 ) and ( 19 ) below , the of a force is defined as the of the average force and the time interval during which the force acts .which one of the following statements is concerning linear momentum ? 1 .momentum is a force .the momentum of an object is always positive .3 .  momentum is a scalar quantity .the si unit of momentum is kgm/s .momentum and impulse have the same units .the brakes of your bicycle have failed , and you must choose between slamming into either a haystack or a concrete wall . which one of the following statements why hitting a haystack is a wiser choice than hitting a concrete wall ? 1 .the haystack gives you a smaller impulse than the concrete wall .the haystack changes your momentum over a longer time .your change in kinetic energy is smaller if you hit the haystack than if you hit the concrete wall .your change in momentum is smaller if you hit the haystack than if you hit the concrete wall .more potential energy is stored in the wall which is released upon the impact and results in a greater force on you .you slide down two consecutive slopes of frictionless ice whose vertical heights are identical , as shown below .select all of the following statements that must bethe change in your kinetic energy is for the motion from a to b and from b to c. 2 .the work done on you by the gravitational force is for the motion from a to b than from b to c. 3 .the work done on you by the gravitational force is for the motion from a to b than from b to c. + 3.4 in 1 .\(1 ) only 2 .\(2 ) only 3 .\(3 ) only 4 .\(1 ) and ( 2 ) only 5 .\(1 ) and ( 3 ) only 21 .rain starts falling vertically down into a cart with frictionless wheels which is moving at a constant velocity on a horizontal surface .the rain drops come to rest with respect to the cart after striking it , and rain water accumulates in the cart .select all of the following statements that must be about this situation .1 .  the cart will continue to move at a constant velocity because the rain is falling vertically while the cart is moving horizontally .2 .  the cart will continue to move at a constant velocity because the total mechanical energy of the cart - rain system is conserved .3 .  the cart will slow down because the horizontal momentum of the cart - rain system is conserved .+ 3.4 in 1 .\(1 ) only 2 .\(2 ) only 3 .\(3 ) only 4 .\(1 ) and ( 2 ) only 5 .  none of the above 22 .three balls are launched from the horizontal level with as shown below .ball ( 1 ) is launched vertically upward , ball ( 2 ) at an angle of , and ball ( 3 ) at an angle of . in order of decreasing speed ( fastest first ) ,rank the each one attains when it reaches the level of the .all three balls have sufficient speed to reach the dashed line . 1 .( 1 ) , ( 2 ) , ( 3 ) 2 .  ( 1 ) , ( 3 ) , ( 2 ) 3 .( 3 ) , ( 2 ) , ( 1 ) 4 .they all have the same speed .5 .  not enough information, their speeds will depend on their masses .you drop two balls of , made of rubber and putty , from the above a horizontal surface ( see figure ) .the rubber ball bounces up after it strikes the surface while the putty ball comes to rest after striking it .assume that in both cases the velocity of the ball takes the to change from its initial to its final value due to contact with the surface .compare the and exerted on the surface by the rubber and putty balls , respectively , .1 .  2 . 3 . 4 . may be smaller or larger than depending upon the relative size of the balls . may be smaller or larger than depending upon the actual height from which the balls are dropped .two blocks are initially at rest on a frictionless horizontal surface .the mass of block a is than the mass of block b. you apply the and pull the blocks through the along a straight line as shown below ( force is applied for the entire distance ) .+ which one of the following statements correctly compares the of the blocks after you pull them the ? 1 .the kinetic energies of both blocks are identical .2 .  the kinetic energy is greater for the smaller mass block because it achieves a larger speed .3 .  the kinetic energy is greater for the larger mass block because of its larger mass .4 .  not enough information , need to know the actual mass of both blocks to compare the kinetic energies . 5 .  not enough information , need to know the actual magnitude of force to compare the kinetic energies . 25 .a box slides with an initial speed on a horizontal surface and eventually comes to a stop .which one of the following is equal to the change in the kinetic energy of the box ? 1 .the momentum of the box multiplied by the distance travelled before coming to rest .2 .  the momentum of the box multiplied by the time elapsed before coming to rest .3 .  the momentum of the box multiplied by the deceleration of the box .4 .  the mass of the box multiplied by the deceleration of the box . 5 .  none of the above .d. hestenes , m. wells , and g. swackhamer , `` force concept inventory , '' phys . teach . *30 * , 141151 ( 1992 ) ; r. k. thornton and d. r. sokoloff , `` assessing student learning of newton s laws : the force and motion conceptual evaluation and the evaluation of active learning laboratory and lecture curricula , '' am .* 66 * , 338352 ( 1998 ) ; r. j. beichner , `` testing student interpretation of kinematics graphs , '' am . j. phys .* 62 * , 750 - 762 ( 1994 ) ; r. thornton , `` questions on heat and temperature , '' unpublished , ( 1992 ) ; d. r. sokoloff , `` electric circuit concept test , '' unpublished , ( 1993 ) ; d. p. maloney , t. l. okuma , c. j. hieggelke , and a. v. heuvelen , `` surveying students conceptual knowledge of electricity and magnetism , '' phys .res . , am . j. phys .* 69(7 ) * , s12-s23 , ( 2001 ) .  a. b. arons , _ a guide to introductory physics teaching _( wiley , ny , 1990 ) ; b. a. sherwood and w. h. bernard , `` work and heat transfer in the presence of sliding friction , '' am . j. phys . * 52 * , 10011007 ( 1984 ) ; a. van heuvelen and x. zou , `` multiple representations of work - energy processes , '' am . j. phys .* 69(2 ) * , 184 - 194 ( 2001 ) ; r. a. lawson and l. c. mcdermott , `` student understanding of the work - energy and impulse - momentum theorems , '' am . j. phys .* 55(9 ) * , 811817 ( 1987 ) ; t. obrien pride , s. vokos , and l. c. mcdermott , `` the challenge of matching learning assessments to teaching goals : an example from the work - energy and impulse - momentum theorems , '' am .* 66 * ( 2 ) , 147157 ( 1998 ) .  c. singh and d. rosengrant , `` students conceptual knowledge of energy and momentum , '' proceedings of the physics education research conference , ( rochester , ny ) , edited by s. franklin , j. marx , and k. cummings , 123 , 2001 .g. aubrecht and j. aubrecht , `` constructing objective tests , '' am .* 51 * , 613620 ( 1983 ) ; a. j. nitko , _ educational assessments of students _( prentice hall / merrill , englewood cliffs , nj , 1996 ) , 2nd ed .m. chi , `` thinking aloud , '' in _ the think aloud method : a practical guide to modeling cognitive processes _ , edited by m. w. van someren , y. f. barnard , and j. a. c. sandberg ( academic press , london , 1994 ) chap.    r. hake , `` interactive - engagement vs. traditional methods : a six - thousand - student survey of mechanics test data for introductory physics courses , '' am .* 66 * , 6474 ( 1998 ) .m. t. h. chi , p. j. feltovich , and r. glaser , categorization and representation of physics knowledge by expert and novices " , cog .sci . * 5 * , 121 - 152 ( 1981 ) .c. singh , when physical intuition fails " , am .j. phys , * 70(11 ) * , 1103 - 1109 , ( 2002 ) ..student performance before and after instruction .the average percent pre-/post - test scores and normalized gain on each item for the upper 25% ( h ) , lower 25% ( l ) , and for all students ( a ) . [ cols="^,^,^,^,^,^,^,^,^,^",options="header " , ]

Abstract: we investigate student understanding of energy and momentum concepts at the level of introductory physics by designing and administering a 25-item multiple choice test and conducting individual interviews . we find that most students have difficulty in qualitatively interpreting basic principles related to energy and momentum and in applying them in physical situations .


electrocardiology rests upon the study of electric fields generated by the heart . as a physicist one immediately asks : what are the sources of these fields .this question has two answers .( i ) on the microscopic scale the sources are and ions and negatively ions and proteins .but there is no large separation of positive and negative charges .the membrane of the active cells is able to open channels for the positive ions only , the negative ones remain confined in the cell , the result is a microscopic dipole .( ii ) on the macroscopic scale , where individual cells can not be resolved , we then have a macroscopic dipole density but no charge density ( monopole ) .the variation of this dipole density in space and time spreads through the tissue as a propagating wave of depolarization .if the dipole density as the source is known it is straightforward to calculate the corresponding electric potential which can be measured .this is the so - called forward problem which in mathematical terms is the solution of poisson s equation .however , the desired medical information is given by the sources , i.e. by the macroscopic dipole density. therefore we must solve the inverse problem of calculating the dipole density from potential measurements . in the pastmost authors have considered a different inverse problem , namely the determination of the potential near the heart wall from potential measurements in the heart chamber or on the body surface . in this caseno assumption about the sources is made , one solves a boundary - value problem for laplace s equation .however , the potential near or on the heart wall is a superposition of the fields generated by near and distant active regions . as a consequence the potential only presents a broad and smooth depiction of the local electric activity .  from these argumentsit is clear that the dipole density is better suited for the mapping of complicated arrhythmias than the potential .one may ask why this approach has not been tried before , to be correct : almost not . there is a pioneering work by a. van oosterom [ 1 ] who always tried to model the sources instead of calculating potentials .he already discusses the scalar dipole density as a model of cardiac activity under the name `` equivalent double layer model '' ( edl ) .the reason why other authors did not follow this route may be the fact that the dipole density is harder to obtain than the contact potential at the heart wall , we shall discuss this in the next section . to compute the dipole density requires much better input data .indeed , four years ago we have made test calculation with data which were collected by the ensite system of st.jude .the results were unsatisfactory . only the new device from acutus medical inc .called acqmap system seems to be suited for our purpose .it has a recording catheter with 48 electrodes plus 48 ultrasound transducers for distance measurements .it allows determination of the heart chamber geometry and the potential measurements simultaneously at the same time .this eliminates the error due to the motion of the measuring electrodes and of the heart itself .the paper is organized as follows . in the next sectionwe discuss the forward and inverse problems for the dipole density .we show that the mapping between the dipole density and the potential is a compact integral operator which can well be represented by a finite dimensional matrix .but its inverse is an unbounded operator .this has the bad consequence that the inverse problem is a so - called ill - defined problem which requires special techniques of solution .these facts are illustrated in section 3 by a simple solvable model which we also use to test the general numerical code for solving the inverse problem for the dipole density . in the last section we consider phase mapping which seems to be the suitable technique to map complicated arrhythmias as atrial fibrillation .we compare the dynamics of phase singularities with the vortex dynamics in 2-dimensional hydrodynamics .we apply macroscopic electrodynamics in matter to the heart filled with blood . at this point oneusually refers to jackson [ 2 ] and starts from the phenomenological maxwell s equations  unfortunately jackson does not make a clear distinction between microscopic electrodynamics in vacuum and macroscopic electrodynamics in matter .so we prefer the concise book by the first author [ 3 ] and refer to the derivation of the macroscopic maxwell s equations given there .since the temporal variation of the cardiac fields is slow compared to the propagation of the fields through the body ( with the speed of light ) , we can use the quasi - static approximation and neglect the time derivatives of and .then by ( 2.1 ) the conduction current is divergence - less and by ( 2.2 ) the electric field is curl - less this implies that the electric field has a scalar potential the blood in the heart is a homogeneous medium with constant conductivity , so that . substituting this into equation ( 2.3 ) yields . then equation ( 2.5 )implies laplace s equation in the quasi - static approximation the time dependence of the fields has completely disappeared .one calculates the potential or dipole density at fixed time , and then makes a movie for successive time instants . summing up we describe the heart as a uniform volume conductor [ 4 ] with electric dipole sources in the heart wall .the laplace equation holds inside the heart chamber filled with blood . in the whole bodythe situation is much more complicated .here we have large inhomogeneities ( lunges and bones ) so that is no longer constant .if one measures potentials on the body surface ( ecg ) one must construct a detailed model of the body , in order to derive the sources from those data . to avoid this severe problemwe assume that we have a multi - electrode catheter in the heart chamber which measures the potential at various locations . in the heart wallwe have the dipole sources which we assume to be localized on a 2-dimensional surface .then instead of ( 2.6 ) we have poisson s equation of the form where is the dirac measure on , is the outer normal at point x on the surface and is the surface dipole density ( dipole strength per area ) .such a source is also called a dipole layer or double layer in electrostatics .the direction of the dipole moment is normal to the surface s. note that this source describes the dipole density .the microscopic dipoles can have different directions , but this microscopic structure can hardly be resolved in detail by macroscopic non - contact techniques .the solution of ( 2.7 ) for is given by the surface integral where is the derivative in the normal direction at point on and is the surface measure .this integral can be rewritten as follows where is the angle between the vector and the normal [ 5 ] .if the dipole density is given the calculation of the potential is straightforward , this is the forward problem .the inverse problem of computing from measured is much harder .the reason is the following .the integral operator ( 2.8 ) which maps to the potential is a operator in the mathematical sense .this important fact must be proved .* proof : *  here we follow the best mathematical reference we know [ 6 ] .let the heart wall be a closed smooth surface where the dipole density is located .this is no serious restriction because at the valves can be put equal to 0 .let be another smooth closed surface completely inside the blood volume where electrodes are placed to measure the potential .the dipole density is assumed to be bounded and continuous on .this implies that is bounded and continuous on , because the kernel in ( 2.9 ) is continuous ( note that and so that we have always ) . as usually let be the banach space of bounded continuous functions on and similarly .then the integral operator ( 2.9 ) maps on and to simplify the notation we write it as this is a bounded operator with the operator norm   to prove that is even compact we must use a decomposition of unity .this is a sequence of positive continuous functions with compact support on with and the following property : for every compact set the intersection of and the support of is not empty for finitely many , only . since the kernel is uniformly continuous on there exists a decomposition of unity and points in the support of such that for all , and arbitrary .the approximating integral operator defined by the sum of product kernels in here is clearly compact .the approximation is in the operator norm because where is the total area of .this proves that is the limit of a converging sequence of compact operators and , therefore , compact .  *end of proof.*.  this fact has one good and one bad consequence .the good one is that compact operators can well be approximated by finite - dimensional matrices . then (2.9 ) becomes a matrix equation the solution of the inverse problem is then given by the inverse matrix compact operators have infinitely many eigenvalues which accumulate only at 0 , for ( theorem of f.riesz [ 6 ] ) . as a consequencethe approximating matrix in ( 2.14 ) has small eigenvalues and this is unavoidable . in the inverse ( 2.15 ) we then have , so that the corresponding part in the data gets strongly amplified .this is the ill - posed nature of the inverse problem . to avoid a huge amplification of the noise one must cut off the smallest eigenvaluesthis is a convenient regularization method called truncated singular value decomposition ( tsvd ) .another widely used method of regularization is the one of tikhonov [ 7 ] . to solve the linear equation ( 2.14 ) one considers the variation principle where is a `` regularizing ' ' operator ( mostly ) and is the regularization parameter . putting the variational derivative equal to 0 we obtain the equation where the cross means the adjoint operator . for positive the accumulation of eigenvalues at 0 is removed in the operator on the left .the latter can be inverted ( the inverse is bounded ) and the dipole density can be computed .the advantage of this regularization method is that the regularization parameter can be varied continuously .the optimal choice of is a serious problem which is discussed in the next section . for onehas the first order tikhonov regularization .the inverse problem of electrocardiology in the standard sense is a voltage to voltage approach where one calculates the potential at points on or near the wall .if the dipole density is known this is a forward calculation ( 2.9 ) with a new integral operator because is now on or near the surface .if is on the wall the kernel of has a singularity at .nevertheless is still compact because it is again the limit of compact operators ( corresponding to a sequence of surfaces converging to from the interior ) . using with the unbounded operator , we can eliminate and get this operator is better behaved than alone , because damps the large eigenvalues of .therefore the standard voltage inverse problem is easier to solve , which means that less strong regularization is necessary .but as discussed in the introduction , it gives less precise information on the electrical activity of the heart .to compare dipole density and potential and test candidate regularization methods , a mathematical model with absolutely known dipole density was defined for which the corresponding potential can be calculated exactly .this model represents dipole density `` frozen '' at one instant of time .a simple solvable model is obtained as follows .let a sphere of radius 1 represent the heart wall ( endocardial surface ) and choose the dipole density applied upon it according to the formula where is the polar angle with respect to the -axis , is a positive parameter and a normalization factor .this distribution is rotationally symmetric around the -axis , it has a maximum at the south pole and diminishes toward the north pole .a large value of the parameter causes the maximum - region on the south pole to be narrow , whereas a small value of causes the maximum - region to be broad .the normalization factor conveniently scales the density values so that the integral over the unit sphere is equal to 1 :   the corresponding voltage for this dipole density can be exactly calculated as follows .we expand in terms of legendre polynomials with respect to the polar angle using the integral where is the modified spherical bessel function .then we obtain with   on the other hand the kernel in the potential integral ( 2.8 ) can also be expanded in terms of legendre polynomials .we start from the well - known expansion where , is the angle between the vectors and and .the normal derivative in ( 2.8 ) on the unit sphere is equal to , hence substituting this into ( 2.8 ) we arrive at a general bounded continuous dipole density on the unit sphere can be expanded in terms of spherical harmonics inserting this into ( 3.8 ) and using the following integral over the angles we get the desired potential in the form this general result can be applied to our rotationally symmetric dipole density ( 3.4 ) which gives the potential everywhere in the unit sphere : with the sum over is rapidly converging so that one can stop at a finite value and gets any desired accuracy .  in figure a we compare and for .we have normalized both quantities to a maximum value 1 for the purpose of direct comparison .as we follow both distributions from maximum ( south pole ) toward the minimum ( north pole ) , the voltage has a long rightward tail ( power law ) , compared to the rapid ( exponential ) descent of the dipole density .this shows clearly the local nature of the dipole density in contrast to the broad distribution of the potential .next we consider the general numerical code .the heart wall is covered by a triangular mesh . since we want to have a continuous dipole density we approximate it by piecewise linear functions and zero otherwise . here , , are the vectors of the corner points of the triangle and det is the determinant of the 3 vectors .these functions satisfy the condition the unknown dipole density is expanded in the form then the integral equation ( 2.8 ) becomes the following set of linear equations for unknowns  where   the integral ( 3.18 ) is a sum of integrals over the triangles with corner .these integrals can be calculated analytically [ 8 ] .the result for one triangle is equal to .\eqno(3.19)\ ] ] here  and is the normal of the triangle and its absolute value where is the area of the triangle .the vector is given by with and cyclic .finally is the solid angle of the triangle subtended at the view point .a convenient formula for has been given by van oosterom and strackee [ 9 ] .if we substitute in ( 3.18 ) by 1 , we obtain the so - called gauss - integral which is equal to the solid angle .since is a closed surface and is in the interior we get .this leads to the sum rule which holds exactly because the discrete triangulated surface subtends the same solid angle .the sum rule is an important test of the code , it must be satisfied with machine accuracy . in other words ,the matrix is a stochastic matrix times , it has an eigenvector with eigenvalue .unfortunately , it also has very small eigenvalues because it approximates a compact operator .then the inverse problem requires regularization .this leads to some error in the resulting dipole density .we now discuss this essential problem in our solvable model .  to be near reality we define a spherical `` basket '' of radius r=0.5 which we first place concentric with the unit sphere representing the heart wall .we calculate the exact potential values at 186 evenly distributed points on this basket .this represents the measured values on an array of electrodes of a basket catheter . finally , we calculate the dipole density on the heart wall ( ) by solving the inverse problem where is a regularized inverse . since we know the exact dipole density we can choose the regularization parameter in an optimal way . using truncated singular - value regularization ( tsvd ) with 110 singular values from total 186we obtain very good results as shown in figure b plotted in red , compared with the exact dipole density values plotted on the blue curve . the normalized rms error is 0.01 . in the case of real data from living heartsa good eye of the medical doctor is required to find out the optimal regularization parameter .if we use tikhonov regularization instead of tsvd we find no statistically significant difference in the resultant calculated dipole density . if the basket is not placed in the center the results get worse , but not dramatically . however using only 48 electrodes instead of186 gives poor results showing that we have a large discretization error in this case .the remedy in view of the real situation with the acqmap system is interpolation of the measured potential values . for this interpolation on a triangular surfacethe method of oostendorp , van oosterom and huiskamp [ 10 ] is very useful , because it minimizes .this is the best strategy because the exact potential would satisfy laplace s equation . .cardiac fibrillation is the main cause of death in the western world .nevertheless its underlying mechanisms of activation are still poorly understood .obviously mapping of cardiac potentials is not sensitive enough to improve the situation .there is considerable hope that dipole density maps can help .these maps show the amplitude of the dipole density distributed over the heart wall ( ) as a function of time .but in addition to the amplitude the phase of the dipole density gives important information as it is the case in the phase analysis of electrograms [ 11 - 14 ] ( and references given there ) .  to define the phase , the dipole density is considered as the real part of a complex function whose imaginary part is given by the hilbert transform where stands for the principle value integral .the phase is then equal to the phase of the complex number , that means if the phase moves out of the interval  ] is reached .this arctan - function is denoted by arctan2 in matlab so that the general definition is in this definition we have assumed that the mean value of over time is zero . byadding or subtracting , can be made continuous in .since the phase can be calculated at every point where the dipole density has been determined , we actually get a phase map on the heart wall for every instant .this map shows singular points where the phase is undetermined .such a phase singularity is actually a singularity of the gradient of .in fact , iif we integrate along a closed curve we get zero , except some singularity of the gradient is included .this is the same situation as in 2-dimensional hydrodynamics where the flow velocity integrated along a closed curve give the circulation which vanishes except a vertex is included . considering as a flow velocity we get a complete hydrodynamical analogy .we have a potential flow , the phase is the velocity potential . in hydrodynamicsthe circulation is conserved in the course of time .we want to investigate the same property for the phase singularities on the heart wall .the first observation is that the phase singularities are quantized vortices .that means the contour integral of ( the circulation ) always has the same value .studying various phase maps on the heart wall we have found that the vortices always appear in pairs : one with circulation and a second one with circulation .this shows that the circulation is indeed conserved like in hydrodynamics .one vortex can not appear or disappear suddenly , it can only annihilate together with a partner of opposite circulation . in the healthy heartthere seems to exist one pair of rather stable vortices only .the two vortices can be joined by a line where the flow velocity is maximal ( see figure 2 ) .this line may be considered as the activation front .the front is most easily found by a jump from to in the phase . during one heartbeat this activation front moves over the heart wall , while its endpoints at the vortices remain more or less fixed .if some arrhytmia is developed , more and more vortex pairs appear and move around . beside vortices sometimessources and perhaps also sinks show up which have a closed activation front . it is clear that the study of this flow dynamics will be an important tool for understanding complicated arrhytmias .   vertices ( black and white ) together with the phase flow .the black line is the activation front .laa : left atrial appendage , lspv : left superior pulmonary vein , lipv : left inferior pulmonary vein , rspv : right superior pulmonary vein , ripv : right inferior pulmonry vein , mv : mitral valve . ]we thank graydon beatty from acutus medical for innumerable elucidating discussions and communication of information .thanks are also due to other members of the acutus team , in particular min zhu and xinwei shi and , of course , randy werneth .kuklik p , zeemering s , maesen b , maesson j , crijns hj , verheule s , ganesan an , scotten u , reconstruction of instantaneous phase of unipolar atrial contact electrogram , ieee trans.biomed .engeneering , 62 , 296 ( 2015 )

Abstract: we discuss the forward and inverse problems between the potential measured in a heart chamber and its sources represented by a dipole density located on the heart wall . we show that the mapping from to is a compact integral operator . its inverse is unbounded which makes the inverse problem ill - posed in the mathematical sense . we investigate methods to solve the inverse problem approximately in view of the mapping of complicated cardiac arrhythmias . we point out an analogy between phase mapping and 2-dimensional hydrodynamics .


[ cols="^ " , ]     ' '' ''  frequency & 60hz ( tuning screws 59.5 to 60.5hz ) +  ' '' ''  moving mass & 2.26 lbs ( 1.03 kg ) +  ' '' ''  total mass & 2.26 lbs ( 1.03 kg ) +  ' '' ''  max stroke ( hard limit ) & 0.135 ( 3.43 mm ) +  ' '' ''  estimated operational stroke & 0.100 ( 2.54 mm ) +  ' '' ''force reacted at 60hz and 0.100 & 334n +  ' '' ''  force reacted at 60hz and 0.135 & 467 n +  fig .[ fig : tva_combined ] shows one of the two gpi cryocoolers with a tva from csa moog mounted .its specifications are listed in tab .[ tab : tva ] .the tva is specified for 60hz and can be finetuned by .5hz using the depicted tuning screw . as stated in sect .[ sect : theory ] the fine tuning is critical for absorbing efficiency . in the worst case amplificationcan occur , particularly if a natural structural mode is located in the close neighborhood .it is notable ( table [ tab : tva ] ) that even though the tva mass is only about one kilogram and has a maximum stroke of a three and a half mm , a reacting force and a `` stroke '' : . ] up to 467n is created at 60hz ( corresponding to 48 kg if translated into weight ) .after shipment and arrival of gpi to cerro pachon in chile , it appeared that the tvas lost their effectiveness which is currently investigated by csa ( and the tvas were dismounted from the cryocooler heads for inspection ) .for the next commissioning run in september 2014 we expect these tvas to be mounted again with a corrected fine - tuning .     ' '' ''  mode & & & & & & +  ' '' ''  & & & & & & +  ' '' ''  1 & interface ring & 2,276 & 48.7 & 22.8 & 0.01 & gpi frame torsion about y +  ' '' ''  2 & interface rings & 2,172 & 52.2 & 21.7 & 0.01 & gpi frame rocking about y +  ' '' ''  3 & cal & 213 & 74.6 & 2.1 & 0.01 & cal plunging in z against rocking ifs +  ' '' ''  4 & cal & 450 & 80.4 & 4.5 & 0.01 & ifs bouncing against gpi frame +  ' '' ''  5 & cal & 690 & 93.8 & 6.9 & 0.01 & cal rocking about y +  ' '' ''  6 & cal & ?& 114.9 & ? & & cal torsion +  ' '' ''  7 & cal & 422 & 126.5 & 4.2 & 0.01 & cal and ifs rocking about x , z +  csa moog was contracted to study gpi s modal frequencies .they prepared a wire frame model for simulation .test measurements were done using a modal hammer to excite the gpi structure and accelerometer sensors were attached to measure the response .[ fig : csa - study - cominbed ] visualizes the wire frame model .the excitation locations are marked with black arrows ; one arrow is hidden behind the purple cal unit .the red dots indicate the locations of the accelerometer sensors . tab .[ tab : gpi - modes ] shows the extracted modes from their data set in the frequency range from 40 to 150hz .all recommended masses for the tmds by csa are based on a mass ratio of .the study was done in december 2012 while gpi was in the a&t laboratory at ucsc and mounted to a carriage creeper ( dubbed the `` l - frame '' ) .the first two modes with the lowest frequencies ( 49 and 52hz ) are likely to shift when gpi is attached to the telescope .this is one of the reasons why no tmds have been installed yet .the decision was taken to revisit this study once enough on - telescope experience is gained to evaluate if tmds are required . to this pointno tmds are installed and so far it seems that no tmds are required .the main vibration line in our error budget ( 60hz ) is not on a structural mode assuming that the two low frequency structural modes of gpi have not shifted through the mounting at the telescope in an unfortunate way .the topic of vibration in general is complex and is a topic prone to misunderstandings between mechanics , opticians , electrical engineers and scientists .an awareness of context and language is critical in this interdisciplinary field . coordinating a sensible and successful vibration mitigation effort in an effective wayis a challenge and requires its resources and roles such as an interconnected project manager and a `` vibration officer '' .there is a constant struggle with hard decisions : does it make sense to spend the extra dollar to finish a study that runs out of funding ? and avoiding the `` sunk cost '' effect in a `` prototype '' development environment is never easy .gpi has the strongest constraints on its wavefront error budget of all gemini south instruments and certainly has risen the observatory s awareness of vibrations in the telescope environment .the mitigation using an lqg controller for both pointing and focus has been implemented successfully .this is an intriguing demonstration of how new technologies can be used to mitigate vibration effects from the controller side . in that sense , using the lqg controller , the impact of the cryocooler vibrations on gpi s performance is low if not negligible .the only exception is the high - order wavefront front sensor of the cal unit .it is a mach - zehnder type interferometer and the loss in fringe contrast due to the vibration prevents its commissioning .fortunately , this is not critical , since the basic functionalities of the cal ( keeping the star bore - sight , and low - order wavefront sensing ) are not impacted .furthermore , the gpi team has developed further alternatives to treat ncpa such as speckle - nulling . and so far , even though other instruments such as gems _ can _ see the vibration signal on their wavefront sensors , it seems that there is no impact on performance .nevertheless , a `` root treatment '' is always preferable and it is in the observatory s natural interest to keep its `` vibration '' spectra as clean as possible .recently , sunpower has developed an active cancellation system ( acs ) as add - on for its gt cryocooler series and we are investigating to upgrade our coolers with it .there is widespread use of tmds for aircrafts , machinery , in the car industry , for trains and railways , buildings , power lines and in optical systems . and certainly there is still plenty of room to make a better use of these devices in the field of astronomical instrumentation and optical telescopes .  a good infrastructure of acceleration sensors is required to effectively understand and solve unwanted vibrations .otherwise measurements are difficult to obtain and to confirm , and resources of the observatory are wasted in large overheads . in the case of the gemini observatory it seems recommendable to have at least 5 sensors around the cell of m1 ( andone could dream about having each hydraulic m1 actuator equipped with a sensor ) .elts will take this matter ( hopefully ) serious , plan for an integrated design and maybe feature a dedicated `` vibration control room '' .as gpi and other ( xao ) instruments can always be seen as pathfinders for future telescopes and instrumentation , sharing our challenges is hopefully helpful to find suitable solutions in other complex environments .gpi can serve as a good example for the successful implementation of a mitigation strategy and for how powerful mitigation strategies such as lgq filtering of the wavefront have become .the authors thank paul langlois for suggestions and improvements on the figures .the gemini observatory is operated by the association of universities for research in astronomy , inc ., under a cooperative agreement with the nsf on behalf of the gemini partnership : the national science foundation ( united states ) , the national research council ( canada ) , conicyt ( chile ) , the australian research council ( australia ) , ministrio da cincia , tecnologia e inovao ( brazil ) and ministerio de ciencia , tecnologa e innovacin productiva ( argentina ).   , b. , graham , j. r. , ingraham , p. , konopacky , q. , marois , c. , perrin , m. , poyneer , l. , bauman , b. , barman , t. , burrows , a. , cardwell , a. , chilcote , j. , de rosa , r. j. , dillon , d. , doyon , r. , dunn , j. , erikson , d. , fitzgerald , m. , gavel , d. , goodsell , s. , hartung , m. , hibon , p. , kalas , p. g. , larkin , j. , maire , j. , marchis , f. , marley , m. , mcbride , j. , millar - 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based and airborne instrumentation for astronomy v _ , _ proc .spie _ * 9147 * ( 2014 ) ., j. k. , larkin , j. e. , maire , j. , perrin , m. d. , fitzgerald , m. p. , doyon , r. , thibault , s. , bauman , b. , macintosh , b. a. , graham , j. r. , and saddlemyer , l. , `` performance of the integral field spectrograph for the gemini planet imager , '' in _ ground - based and airborne instrumentation for astronomy iv _ , _ proc .spie _ * 8446 * ( sept . 2012 ) .poyneer , l. a. , de rosa , r. j. , macintosh , b. , palmer , d. w. , perrin , m. d. , sadakuni , n. , savransky , d. , bauman , b. , cardwell , a. , chilcote , j. k. , dillon , d. , gavel , d. , goodsell , s. j. , hartung , m. , hibon , p. , rantakyr , f. t. , thomas , s. , and vran , j .-, `` on - sky performance during verification and commissioning of the gemini planet imager s adaptive optics system , '' in _ adaptive optics systems iv _ , _ proc .spie * 9148 * _ ( 2014 ) .sivo , g. , kulcsr , c. , conan , j .-m . , raynaud , h .- f . ,gendron , e. , basden , a. , vidal , f. , morris , t. , meimon , s. , petit , c. , gratadour , d. , martin , o. , hubert , z. , sevin , a. , perret , d. , chemla , f. , rousset , g. , dipper , n. , talbot , g. , younger , e. , myers , r. , henry , d. , todd , s. , atkinson , d. , dickson , c. , and langmore , a. , `` first on - 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Abstract: the gemini planet imager ( gpi ) entered on - sky commissioning and had its first - light at the gemini south ( gs ) telescope in november 2013 . gpi is an extreme adaptive optics ( xao ) , high - contrast imager and integral - field spectrograph dedicated to the direct detection of hot exo - planets down to a jupiter mass . the performance of the apodized pupil lyot coronagraph depends critically upon the residual wavefront error ( design goal of 60nmrms with rms tip / tilt ) , and therefore is most sensitive to vibration ( internal or external ) of gemini s instrument suite . excess vibration can be mitigated by a variety of methods such as passive or active dampening at the instrument or telescope structure or kalman filtering of specific frequencies with the ao control loop . understanding the sources , magnitudes and impact of vibration is key to mitigation . this paper gives an overview of related investigations based on instrument data ( gpi ao module ) as well as external data from accelerometer sensors placed at different locations on the gs telescope structure . we report the status of related mitigation efforts , and present corresponding results .   [ sec : intro ]