# Kinematics

## Acceleration


### Learning Objectives

By the end of this section, you will be able to:
1. Define and distinguish between instantaneous acceleration, average acceleration, and deceleration.
2. Calculate acceleration given initial time, initial velocity, final time, and final velocity.
In everyday conversation, to accelerate means to speed up. The accelerator in a car can in fact cause it to speed up. The greater the acceleration, the greater the change in velocity over a given time. The formal definition of acceleration is consistent with these notions, but more inclusive.
Because acceleration is velocity in m/s divided by time in s, the SI units for acceleration are , meters per second squared or meters per second per second, which literally means by how many meters per second the velocity changes every second.
Recall that velocity is a vector—it has both magnitude and direction. This means that a change in velocity can be a change in magnitude (or speed), but it can also be a change in direction. For example, if a car turns a corner at constant speed, it is accelerating because its direction is changing. The quicker you turn, the greater the acceleration. So there is an acceleration when velocity changes either in magnitude (an increase or decrease in speed) or in direction, or both.
Keep in mind that although acceleration is in the direction of the change in velocity, it is not always in the direction of motion. When an object slows down, its acceleration is opposite to the direction of its motion. This is known as deceleration. 

### Instantaneous Acceleration

Instantaneous acceleration , or the acceleration at a specific instant in time, is obtained by the same process as discussed for instantaneous velocity in Time, Velocity, and Speed—that is, by considering an infinitesimally small interval of time. How do we find instantaneous acceleration using only algebra? The answer is that we choose an average acceleration that is representative of the motion.  shows graphs of instantaneous acceleration versus time for two very different motions. In (a), the acceleration varies slightly and the average over the entire interval is nearly the same as the instantaneous acceleration at any time. In this case, we should treat this motion as if it had a constant acceleration equal to the average (in this case about ). In (b), the acceleration varies drastically over time. In such situations it is best to consider smaller time intervals and choose an average acceleration for each. For example, we could consider motion over the time intervals from 0 to 1.0 s and from 1.0 to 3.0 s as separate motions with accelerations of  and , respectively.
The next several examples consider the motion of the subway train shown in . In (a) the shuttle moves to the right, and in (b) it moves to the left. The examples are designed to further illustrate aspects of motion and to illustrate some of the reasoning that goes into solving problems.
The graphs of position, velocity, and acceleration vs. time for the trains in  and  are displayed in . (We have taken the velocity to remain constant from 20 to 40 s, after which the train decelerates.)

### Sign and Direction

Perhaps the most important thing to note about these examples is the signs of the answers. In our chosen coordinate system, plus means the quantity is to the right and minus means it is to the left. This is easy to imagine for displacement and velocity. But it is a little less obvious for acceleration. Most people interpret negative acceleration as the slowing of an object. This was not the case in , where a positive acceleration slowed a negative velocity. The crucial distinction was that the acceleration was in the opposite direction from the velocity. In fact, a negative acceleration will increase a negative velocity. For example, the train moving to the left in  is sped up by an acceleration to the left. In that case, both  and  are negative. The plus and minus signs give the directions of the accelerations. If acceleration has the same sign as the velocity, the object is speeding up. If acceleration has the opposite sign as the velocity, the object is slowing down. 

### Test Prep for AP Courses


### Section Summary

1. Acceleration is the rate at which velocity changes. In symbols, average acceleration  is
      
2. The SI unit for acceleration is .
3. Acceleration is a vector, and thus has a both a magnitude and direction. 
4. Acceleration can be caused by either a change in the magnitude or the direction of the velocity.
5. Instantaneous acceleration  is the acceleration at a specific instant in time. 
6. Deceleration is an acceleration with a direction opposite to that of the velocity.

### Conceptual Questions


### Problems & Exercises
