Practice Problem Set: Motion Along a Line

How to Use This Problem Set

Motion along a line is one of the most important applications of derivatives and integrals in AP Calculus. A particle's position tells you where it is, its velocity tells you how its position is changing, and its acceleration tells you how its velocity is changing.

Before working the problems, keep the fundamental relationships in mind:

The most important distinction is between displacement and total distance traveled. Displacement measures the net change in position. Total distance counts movement in both directions and therefore requires you to account for changes in direction.

Core Motion Relationships

Position, Velocity, and Acceleration

If the position of a particle at time is , then velocity is the derivative of position and acceleration is the derivative of velocity:

The sign of velocity tells you the particle's direction of motion:

If , the particle is moving in the positive direction.
If , the particle is moving in the negative direction.
If , the particle is momentarily at rest.

Speed

Speed is the magnitude of velocity, so:

Speed is never negative. A particle can have negative velocity while still having positive speed.

Displacement

If a particle moves from time to time , its displacement is:

Displacement can be positive, negative, or zero.

Total Distance Traveled

Total distance is different. If velocity changes sign during the interval, the particle changes direction, and you must break the interval into pieces. In integral form:

For AP Calculus problems, a reliable strategy is to find all times when velocity is zero, determine where the particle changes direction, and then calculate the distance traveled on each resulting interval.

Practice Problems

Question 1 — Position and Velocity

A particle moves along the -axis with position , where is measured in seconds and position is measured in meters. Find the particle's velocity at .

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Question 2 — Acceleration

A particle has velocity . Find its acceleration at time .

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Question 3 — Direction of Motion

A particle's velocity is given by . On which intervals in is the particle moving to the right?

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Question 4 — Speed

A particle has velocity . Find the speed of the particle at .

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Question 5 — When Is the Particle at Rest?

A particle moves with velocity for . Find all times when the particle is at rest.

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Question 6 — Displacement

A particle has velocity . Find the displacement of the particle from to .

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Question 7 — Total Distance Traveled

A particle has velocity on the interval . Find the total distance traveled by the particle.

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Question 8 — Combined Motion Analysis

A particle has position . Determine the particle's velocity and acceleration at , and state whether the particle is moving to the right or left at that time.

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Question 9 — Direction Changes

A particle has velocity on . Find the times when the particle changes direction.

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Question 10 — Displacement vs. Distance

A particle has velocity for . Find both the displacement and the total distance traveled.

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Worked Example: Finding Total Distance

Example: A Particle That Changes Direction

Suppose the velocity of a particle is for . Find the total distance traveled.

Step 1: Find when the particle is at rest:

Step 2: Solve for the critical times:

Step 3: Determine the sign of velocity on each interval. The velocity is positive before the first zero and negative between the two zeros, so the particle changes direction at each critical time.

Step 4: Add the distances traveled on each interval:

Result:

Common Mistakes

Confusing velocity with speed

Velocity includes direction and can be negative. Speed is the magnitude of velocity and is always nonnegative. If the question asks for speed, use .

Using displacement when the question asks for distance

Displacement is the net change in position. Total distance counts all movement, including movement in both directions. If velocity changes sign, you generally need to split the interval before calculating total distance.

Forgetting that velocity zero does not always mean a direction change

A particle is at rest whenever velocity equals zero, but it changes direction only if velocity changes sign. Always check the sign of velocity on both sides of the critical time.

Confusing acceleration with speeding up

A particle is speeding up when velocity and acceleration have the same sign. It is slowing down when velocity and acceleration have opposite signs. The sign of acceleration by itself does not tell you whether the particle is speeding up or slowing down.

AP Calculus Motion Checklist

When you encounter a motion-along-a-line problem, work through this checklist:

Identify the position function if one is given.
Differentiate position to get velocity.
Differentiate velocity to get acceleration.
Set velocity equal to zero to find possible times when the particle is at rest.
Check the sign of velocity to determine direction of motion.
Use absolute velocity for speed.
Use the net change in position for displacement.
Split the interval at direction changes when calculating total distance.

If you can move comfortably between these relationships, you can handle most of the standard motion problems appearing in AP Calculus AB and BC.

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