# Acceleration

## Representing Acceleration with Equations and Graphs


### Section Key Terms


### How the Kinematic Equations are Related to Acceleration

We are studying concepts related to motion: time, displacement, velocity, and especially acceleration. We are only concerned with motion in one dimension. The kinematic equations that we will be using apply to conditions of constant acceleration, except where noted, and show how these concepts are related. Constant acceleration is acceleration that does not change over time. The first kinematic equation relates displacement d, average velocity  
, and time t.
 
The initial displacement  

 is often 0, in which case the equation can be written as  

This equation, which is the definition of average velocity and valid for both constant and non-constant acceleration, says that average velocity is displacement per unit time. We will express velocity in meters per second. If we graph displacement versus time, as in , the slope will be the velocity. Whenever a rate, such as velocity, is represented graphically, time is usually taken to be the independent variable and is plotted along the x axis.
The second kinematic equation, another expression for average velocity  
 
  
 is simply the initial velocity plus the final velocity divided by two. This equation is only valid for constant acceleration.
Now we come to our main focus of this chapter; namely, the kinematic equations that describe motion with constant acceleration. In the third kinematic equation, acceleration is the rate at which velocity increases, so velocity at any point equals initial velocity plus acceleration multiplied by time
Note that this third kinematic equation does not have displacement in it. Therefore, if you do not know the displacement and are not trying to solve for a displacement, this equation might be a good one to use.
The third kinematic equation is also represented by the graph in .
The fourth kinematic equation shows how displacement is related to acceleration
When starting at the origin,    and, when starting from rest,  , in which case the equation can be written as
This equation tells us that, for constant acceleration, the slope of a plot of 2d versus t2 is acceleration, as shown in .
The fifth kinematic equation relates velocity, acceleration, and displacement
This equation is useful for when we do not know, or do not need to know, the time.
When starting from rest, the fifth equation simplifies to
According to this equation, a graph of velocity squared versus twice the displacement will have a slope equal to acceleration.
Note that, in reality, knowns and unknowns will vary. Sometimes you will want to rearrange a kinematic equation so that the knowns are the values on the axes and the unknown is the slope. Sometimes the intercept will not be at the origin (0,0). This will happen when v0 or d0 is not zero. This will be the case when the object of interest is already in motion, or the motion begins at some point other than at the origin of the coordinate system.
The kinematic equations are applicable when you have constant acceleration.
1.  
 , or  
   when d0 = 0
2.  
  
3.  
 , or  
   when v0 = 0
4.  
 , or  
   when d0 = 0 and v0 = 0
5.  
 , or  
   when d0 = 0 and v0 = 0

### Applying Kinematic Equations to Situations of Constant Acceleration

Problem-solving skills are essential to success in a science and life in general. The ability to apply broad physical principles, which are often represented by equations, to specific situations is a very powerful form of knowledge. It is much more powerful than memorizing a list of facts. Analytical skills and problem-solving abilities can be applied to new situations, whereas a list of facts cannot be made long enough to contain every possible circumstance. Essential analytical skills will be developed by solving problems in this text and will be useful for understanding physics and science in general throughout your life.

### Problem-Solving Steps

While no single step-by-step method works for every problem, the following general procedures facilitate problem solving and make the answers more meaningful. A certain amount of creativity and insight are required as well.
1. Examine the situation to determine which physical principles are involved. It is vital to draw a simple sketch at the outset. Decide which direction is positive and note that on your sketch.
2. Identify the knowns: Make a list of what information is given or can be inferred from the problem statement. Remember, not all given information will be in the form of a number with units in the problem. If something starts from rest, we know the initial velocity is zero. If something stops, we know the final velocity is zero.
3. Identify the unknowns: Decide exactly what needs to be determined in the problem.
4. Find an equation or set of equations that can help you solve the problem. Your list of knowns and unknowns can help here. For example, if time is not needed or not given, then the fifth kinematic equation, which does not include time, could be useful.
5. Insert the knowns along with their units into the appropriate equation and obtain numerical solutions complete with units. This step produces the numerical answer; it also provides a check on units that can help you find errors. If the units of the answer are incorrect, then an error has been made.
6. Check the answer to see if it is reasonable: Does it make sense? This final step is extremely important because the goal of physics is to accurately describe nature. To see if the answer is reasonable, check its magnitude, its sign, and its units. Are the significant figures correct?

### Summary of Problem Solving

1. Determine the knowns and unknowns.
2. Find an equation that expresses the unknown in terms of the knowns. More than one unknown means more than one equation is needed.
3. Solve the equation or equations.
4. Be sure units and significant figures are correct.
5. Check whether the answer is reasonable.

### Practice Problems


### Constant Acceleration

In many cases, acceleration is not uniform because the force acting on the accelerating object is not constant over time. A situation that gives constant acceleration is the acceleration of falling objects. When air resistance is not a factor, all objects near Earth’s surface fall with an acceleration of about 9.80 m/s2. Although this value decreases slightly with increasing altitude, it may be assumed to be essentially constant. The value of 9.80 m/s2 is labeled g and is referred to as acceleration due to gravity. Gravity is the force that causes nonsupported objects to accelerate downward—or, more precisely, toward the center of Earth. The magnitude of this force is called the weight of the object and is given by mg where m is the mass of the object (in kg). In places other than on Earth, such as the Moon or on other planets, g is not 9.80 m/s2, but takes on other values. When using g for the acceleration a in a kinematic equation, it is usually given a negative sign because the acceleration due to gravity is downward.

### Practice Problems


### Check Your Understanding


### Section Summary

1. The kinematic equations show how time, displacement, velocity, and acceleration are related for objects in motion.
2. In general, kinematic problems can be solved by identifying the kinematic equation that expresses the unknown in terms of the knowns.
3. Displacement, velocity, and acceleration may be displayed graphically versus time.
4. The kinematic equations in this section are valid only for constant acceleration.

### Key Equations


### Concept Items


### Critical Thinking


### Problems


### Performance Task


### Test Prep Multiple Choice


### Test Prep Short Answer


### Test Prep Extended Response
