The Big Idea
Motion problems are one of the most important applications of derivatives in AP Calculus. The key is to understand that position, velocity, and acceleration are different functions describing different aspects of the same motion.
If an object's position at time is , then velocity is the derivative of position, and acceleration is the derivative of velocity.
So the three functions form a chain:
If you are given position, differentiate once to get velocity and differentiate twice to get acceleration. If you are given acceleration, you can work backward using integration—but you will need initial conditions to determine the constants.
Position
The position function tells you where an object is located at a particular time.
We usually write position as , where represents time. Depending on the problem, position might be measured in feet, meters, miles, or another unit.
For example, if , then at time the object is located at position .
Velocity
Velocity measures how quickly position is changing with respect to time. Because it is the derivative of position:
Velocity has both magnitude and direction. A positive velocity means the object's position is increasing; a negative velocity means the object's position is decreasing.
Do not confuse position with velocity. Position tells you where the object is. Velocity tells you how its position is changing.
Acceleration
Acceleration measures how quickly velocity is changing. Therefore, acceleration is the derivative of velocity:
Because velocity is itself the derivative of position, acceleration is the second derivative of position:
This means that if you are given a position function, you can find everything else by differentiating.
Worked Example 1: Find Position, Velocity, and Acceleration
An object moves along a straight line with position function , where is measured in seconds and position is measured in meters. Find the velocity and acceleration at time .
What Does the Sign of Velocity Mean?
The sign of velocity tells you the direction in which the object's position is changing.
- If : the object is moving in the positive direction.
- If : the object is moving in the negative direction.
- If : the object is momentarily at rest.
Notice that "positive" and "negative" are about direction, not necessarily about whether the object is speeding up or slowing down.
Speeding Up vs. Slowing Down
This is one of the most frequently tested ideas in AP Calculus motion problems.
An object is speeding up when velocity and acceleration have the same sign. It is slowing down when velocity and acceleration have opposite signs.
The easiest way to remember this is to make a sign chart for and .
Suppose at a particular time an object has velocity and acceleration . Is the object speeding up or slowing down?
At another time, suppose and . Is the object speeding up or slowing down?
When Does the Object Change Direction?
An object changes direction when its velocity changes from positive to negative or from negative to positive.
That means you should look for points where:
But there is an important detail: velocity equal to zero does not automatically mean that the object changes direction. You must check whether the sign of velocity actually changes.
Suppose . Determine when the object changes direction.
Average Velocity vs. Instantaneous Velocity
AP Calculus often asks you to distinguish between average velocity over an interval and instantaneous velocity at a specific time.
Average velocity is the change in position divided by the change in time:
Instantaneous velocity is the derivative of position:
Geometrically, average velocity is the slope of a secant line through the position function, while instantaneous velocity is the slope of the tangent line.
An object's position is . Find its average velocity on the interval .
Distance Traveled vs. Displacement
Another important distinction is between displacement and total distance traveled.
Displacement is simply the change in position:
Total distance traveled, however, must account for movement in both directions. If velocity changes sign, you must split the interval at those times and add the absolute changes in position.
To find total distance traveled from a position function, first find where . Those are the times when the object may change direction. Then evaluate the position at all relevant endpoints and critical times, and add the absolute differences.
Suppose an object has position on . Find the total distance traveled.
Working Backward: From Acceleration to Position
So far, we have differentiated to move from position to velocity to acceleration. But AP Calculus also asks you to work backward.
If acceleration is known, integrate once to obtain velocity:
Then integrate velocity to obtain position:
When integrating, remember the constant of integration. An initial condition such as allows you to determine that constant.
An object has acceleration and initial velocity . Find its velocity function.
A Reliable Motion-Problem Strategy
When you see a particle-motion problem on the AP exam, use this sequence.
- Identify what function you were given. Is it position, velocity, or acceleration?
- Move between functions correctly. Differentiate to move forward; integrate to move backward.
- Keep track of signs. Positive and negative velocity indicate direction.
- For speeding up/slowing down, compare signs of velocity and acceleration.
- For direction changes, check whether velocity changes sign.
- For total distance, split the interval wherever velocity changes sign.
- Include units. Velocity has units of distance/time; acceleration has units of distance/time squared.
Common Mistakes
Acceleration by itself does not tell you whether an object is speeding up. You must compare the signs of velocity and acceleration. A negative velocity combined with positive acceleration means the object is slowing down.
An object can have velocity zero without changing direction. Always check the sign of velocity immediately before and after the time.
Displacement can cancel movement in opposite directions. Total distance cannot. If the particle changes direction, split the interval and add the absolute distances traveled.
When you integrate acceleration to obtain velocity, or velocity to obtain position, include the constant of integration and use the given initial condition to determine it.
Practice Problems
A particle has position . Find its velocity and acceleration functions.
Show answer
A particle has velocity . At what times is the particle moving to the right?
Show answer
At a particular time, a particle has velocity and acceleration . Is it speeding up or slowing down?
Show answer
A particle has velocity . At which times does the particle change direction?
Show answer
A particle has position . Find its average velocity on .
Show answer
A particle has position on . Find the total distance traveled.
Show answer
Test yourself: AP Calculus practice exams
Ready to check what you know? Work an AP-style practice exam, then open the full solutions.
Want all 1,114 practice problems?
This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.
Get the Calc Pass App
Scan to download and practice on the go, with instant scoring.
Vol. 1: 1,114 Questions
The complete practice question bank, organized by AB & BC topic.
View on Amazon