Using Integrals to Model Particle Motion

The Big Idea: Motion Is Connected by Derivatives and Integrals

Particle-motion problems become much easier once you see the relationship among position, velocity, and acceleration. Each quantity is a derivative of the one before it, and integration reverses those relationships.

If you know a particle's velocity, integrating velocity gives you its change in position. If you know its acceleration, integrating acceleration gives you its change in velocity.

The key is to pay attention to what the problem actually asks for. Displacement, position, and total distance are related, but they are not the same thing.

Position, Velocity, and Acceleration

Suppose the position of a particle at time is . Its velocity and acceleration are obtained by differentiating:

Velocity tells you the particle's instantaneous rate of change of position. Its sign tells you the direction of motion:

• If , the particle is moving to the right.
• If , the particle is moving to the left.
• If , the particle is momentarily at rest.

Acceleration describes how velocity is changing:

Using an Integral to Find Displacement

Because velocity is the derivative of position, integrating velocity over a time interval gives the particle's displacement during that interval.

Displacement is the net change in position. A positive displacement means the final position is greater than the initial position; a negative displacement means the final position is smaller.

Worked Example 1: Find Displacement from Velocity

Displacement Is Not Total Distance

This distinction is one of the most important ideas in AP Calculus particle-motion problems.

Displacement allows positive and negative motion to cancel. Total distance does not.

To find total distance, you therefore need to determine where the velocity changes sign. Those points divide the time interval into pieces on which the particle moves in a single direction.

Worked Example 2: Total Distance Traveled

Recovering Position from Velocity

Sometimes a problem gives you velocity but asks for the particle's position at a later time. An integral gives the change in position, but you also need the initial position.

This is the Fundamental Theorem of Calculus in action: the integral of velocity from to gives the change from the initial position to the position at time .

Worked Example 3: Find Position from Velocity

Using Acceleration to Find Velocity

The exact same idea works one level higher. Since acceleration is the derivative of velocity, integrating acceleration gives the change in velocity.

If the problem gives an initial velocity, you can use it to determine the velocity at any later time.

Worked Example 4: Acceleration to Velocity

When Does the Particle Change Direction?

A particle changes direction when its velocity changes sign. The first step is to find candidate times by solving:

But finding a zero of velocity is not enough. A particle changes direction only if velocity actually changes from positive to negative or from negative to positive.

For example, if velocity is positive on both sides of a time where , the particle momentarily stops but does not reverse direction.

Common Mistakes

A Reliable AP Exam Strategy

1. Identify what is given: position, velocity, or acceleration.
2. Identify what is being asked: position, velocity, displacement, distance, or direction of motion.
3. Use derivatives to move from position to velocity to acceleration.
4. Use integrals to move backward: acceleration to velocity, or velocity to position.
5. For displacement: integrate velocity directly.
6. For total distance: find where velocity changes sign, then integrate speed on each interval.
7. Interpret the answer with units and context.

The Three Formulas to Know

If you remember these relationships, most particle-motion problems become a matter of identifying the correct quantity and applying the appropriate integral.

Practice Problems