AP Calculus AB and BC · Exam guide
AP Calculus Particle Motion FRQ: Position, Velocity, Acceleration, and Speed
Particle motion questions ask about the same four ideas every time: where the particle is, which way it moves, how fast, and how far it travels in total. Each idea is tied to a rule, and graders give points for writing the rule. This guide works one example.
The four ideas
| Question | Rule |
|---|---|
| Where is the particle? | Position \(x(t) = x(0) + \int_0^t v(s)\,ds\). |
| Which way is it moving? | Right if \(v(t) \gt 0\), left if \(v(t) \lt 0\), and at rest only when \(v(t) = 0\). |
| Is the speed increasing? | Speed increases when \(v\) and \(a\) have the same sign, and decreases when they have opposite signs. |
| How far has it traveled? | Total distance is \(\int |v(t)|\,dt\). Net change in position is \(\int v(t)\,dt\). |
A worked example
A particle moves along a line with velocity \(v(t) = t^2 - 4t + 3\) for \(0 \le t \le 4\), and \(x(0) = 1\). (a) When is the particle moving to the left? (b) Find the position at \(t = 4\). (c) Find the total distance traveled in the first 4 seconds. (d) Is the speed increasing or decreasing at \(t = 2.5\)?
(a) \(v(t) = (t - 1)(t - 3)\). \(v \lt 0\) on \((1, 3)\), so the particle moves left for \(1 \lt t \lt 3\).
(b) \[x(4) = 1 + \int_0^4 (t^2 - 4t + 3)\,dt = 1 + \left[\dfrac{t^3}{3} - 2t^2 + 3t\right]_0^4 = 1 + \dfrac{4}{3} = \dfrac{7}{3} \approx 2.333\]
(c) The velocity changes sign at \(t = 1\) and \(t = 3\). \(\int_0^4 |v|\,dt = \dfrac{4}{3} + \dfrac{4}{3} + \dfrac{4}{3} = 4\).
(d) \(v(2.5) = -0.75\) and \(a(t) = 2t - 4\), so \(a(2.5) = 1\). The signs are opposite, so the speed is decreasing.
It uses only the sign of the acceleration, not the sign of the velocity.
It gives both values and the rule.
Why total distance is different from net change
In part (b) the net change in position is 4/3, but in part (c) the total distance is 4, because the particle goes right, then left by the same amount, then right again. Whenever velocity changes sign, split the integral at those times before taking absolute values.
Common mistakes
- Using the sign of acceleration alone to decide whether the speed is increasing.
- Using \(\int v\,dt\) for total distance when velocity changes sign.
- Saying the particle is at rest when acceleration is zero.
- Leaving out the initial position when finding a position.
Frequently asked questions
What is the difference between speed and velocity?
Velocity has a sign, giving direction. Speed is the absolute value of velocity.
When is a particle at rest?
When the velocity is zero.
How do I find a change of direction?
Find where the velocity changes sign, not just where it equals zero.
Is this the same in AB and BC?
The one-dimensional case is. BC also includes particles in the plane, using parametric or vector functions.
Where can I find the official scoring rules?
Exam formats change a little from year to year, so check the current course and exam description on the College Board site before test day.
Keep practicing
Every lesson in the AP Calculus courses ends with practice problems and an Exam focus box that shows how the topic is tested, and every unit has a 10-problem practice set with worked answers. These free lessons also build the skills the free-response section tests:
See the AP Calculus AB and BC course Free lessons and lesson list See the AP Calculus BC course
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Multiple choice
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