# Kinematics

## Motion Equations for Constant Acceleration in One Dimension


### Learning Objectives

By the end of this section, you will be able to:
1. Calculate displacement of an object that is not accelerating, given initial position and velocity.
2. Calculate final velocity of an accelerating object, given initial velocity, acceleration, and time.
3. Calculate displacement and final position of an accelerating object, given initial position, initial velocity, time, and acceleration.
We might know that the greater the acceleration of, say, a car moving away from a stop sign, the greater the displacement in a given time. But we have not developed a specific equation that relates acceleration and displacement. In this section, we develop some convenient equations for kinematic relationships, starting from the definitions of displacement, velocity, and acceleration already covered.

### Notation: t, x, v, a

First, let us make some simplifications in notation. Taking the initial time to be zero, as if time is measured with a stopwatch, is a great simplification. Since elapsed time is , taking  means that , the final time on the stopwatch. When initial time is taken to be zero, we use the subscript 0 to denote initial values of position and velocity. That is,  is the initial position  and  is the initial velocity. We put no subscripts on the final values. That is,   is the final time,   is the final position, and  is the final velocity. This gives a simpler expression for elapsed time—now, . It also simplifies the expression for displacement, which is now . Also, it simplifies the expression for change in velocity, which is now . To summarize, using the simplified notation, with the initial time taken to be zero,
where the subscript 0 denotes an initial value and the absence of a subscript denotes a final value in whatever motion is under consideration.
We now make the important assumption that acceleration is constant. This assumption allows us to avoid using calculus to find instantaneous acceleration. Since acceleration is constant, the average and instantaneous accelerations are equal. That is,
so we use the symbol  for acceleration at all times. Assuming acceleration to be constant does not seriously limit the situations we can study nor degrade the accuracy of our treatment. For one thing, acceleration is constant in a great number of situations. Furthermore, in many other situations we can accurately describe motion by assuming a constant acceleration equal to the average acceleration for that motion. Finally, in motions where acceleration changes drastically, such as a car accelerating to top speed and then braking to a stop, the motion can be considered in separate parts, each of which has its own constant acceleration.
The equation  reflects the fact that, when acceleration is constant,  is just the simple average of the initial and final velocities. For example, if you steadily increase your velocity (that is, with constant acceleration) from 30 to 60 km/h, then your average velocity during this steady increase is 45 km/h. Using the equation  to check this, we see that
which seems logical.
The equation  gives insight into the relationship between displacement, average velocity, and time. It shows, for example, that displacement is a linear function of average velocity. (By linear function, we mean that displacement depends on  rather than on  raised to some other power, such as . When graphed, linear functions look like straight lines with a constant slope.) On a car trip, for example, we will get twice as far in a given time if we average 90 km/h than if we average 45 km/h.
In addition to being useful in problem solving, the equation  gives us insight into the relationships among velocity, acceleration, and time. From it we can see, for example, that 
1. final velocity depends on how large the acceleration is and how long it lasts
2. if the acceleration is zero, then the final velocity equals the initial velocity , as expected (i.e., velocity is constant)
3. if  is negative, then the final velocity is less than the initial velocity
(All of these observations fit our intuition, and it is always useful to examine basic equations in light of our intuition and experiences to check that they do indeed describe nature accurately.)
What else can we learn by examining the equation  We see that:
1. displacement depends on the square of the elapsed time when acceleration is not zero. In , the dragster covers only one fourth of the total distance in the first half of the elapsed time
2. if acceleration is zero, then the initial velocity equals average velocity () and  becomes 
An examination of the equation  can produce further insights into the general relationships among physical quantities:
1. The final velocity depends on how large the acceleration is and the distance over which it acts
2. For a fixed deceleration, a car that is going twice as fast doesn’t simply stop in twice the distance—it takes much further to stop. (This is why we have reduced speed zones near schools.) 

### Putting Equations Together

In the following examples, we further explore one-dimensional motion, but in situations requiring slightly more algebraic manipulation. The examples also give insight into problem-solving techniques. The box below provides easy reference to the equations needed. 
With the basics of kinematics established, we can go on to many other interesting examples and applications. In the process of developing kinematics, we have also glimpsed a general approach to problem solving that produces both correct answers and insights into physical relationships. Problem-Solving Basics discusses problem-solving basics and outlines an approach that will help you succeed in this invaluable task.

### Test Prep for AP Courses


### Section Summary

1. To simplify calculations we take acceleration to be constant, so that  at all times. 
2. We also take initial time to be zero. 
3. Initial position and velocity are given a subscript 0; final values have no subscript. Thus,
    
4. The following kinematic equations for motion with constant  are useful:
    
      
    
      
    
      
    
5. In vertical motion,  is substituted for .

### Problems & Exercises
