Calculus AB · Unit 4: Contextual Applications of Differentiation · 14 min read · Updated 2026-05-10
Straight-line motion: position, velocity, acceleration — AP Calculus AB
AP Calculus AB · Unit 4: Contextual Applications of Differentiation · 14 min read
1. Position and Instantaneous Velocity★★☆☆☆⏱ 4 min
Position of a particle at time $t$ is given by the function $s(t)$, which returns the coordinate of the particle on the straight axis. Time is always non-negative ($t \geq 0$) in AP motion problems, and displacement between $t=a$ and $t=b$ is $s(b)-s(a)$, the net change in position, distinct from total distance traveled.
v(t) = s'(t) = \frac{ds}{dt}
The sign of velocity tells you direction of motion: $v(t) > 0$ means the particle moves in the positive direction (right/up), $v(t) < 0$ means it moves in the negative direction (left/down), and $v(t)=0$ means the particle is instantaneously at rest. Common exam questions ask for velocity at a specific time or for all times when the particle is at rest.
Exam tip: Always include units in your final answer for FRQ motion problems. AP graders dock 1 point for missing or incorrect units on contextual questions, so add units to every final answer by default.
2. Acceleration and Speeding Up/Slowing Down★★☆☆☆⏱ 3 min
Acceleration is the instantaneous rate of change of velocity with respect to time, making it the second derivative of the original position function.
A key conceptual skill tested heavily on the AP exam is determining if a particle is speeding up or slowing down. The rule is: if $v(t)$ and $a(t)$ have the same sign (both positive or both negative), the particle is speeding up; if they have opposite signs, the particle is slowing down.
Exam tip: Never assume that negative acceleration means the particle is slowing down. The sign of acceleration alone only describes how velocity is changing; you always need to compare the sign of acceleration to the sign of velocity to check if speed is increasing or decreasing.
3. Speed and its Rate of Change★★★☆☆⏱ 4 min
Velocity and speed are often confused, but they are distinct quantities. Velocity is a one-dimensional vector that includes direction encoded in its sign. Speed is a non-negative scalar that has no direction, equal to the absolute value of velocity.
\text{Speed}(t) = |v(t)|
AP Calculus AB also tests the rate of change of speed, which can be derived using the chain rule for absolute value, for $v(t) \neq 0$:
The sign of this derivative confirms the earlier speeding up/slowing down rule: when $v$ and $a$ have the same sign, the derivative of speed is positive (speed increasing), and when opposite, it is negative (speed decreasing).
Exam tip: Circle the word "speed" every time it appears in a question, to remind yourself to take the absolute value of velocity. It is one of the most common avoidable mistakes on AP motion questions.
4. AP-Style Practice Worked Examples★★★☆☆⏱ 3 min
Common Pitfalls
Why: Students memorize "particle changes direction when velocity is zero" but forget that velocity must change sign at that point; a particle can stop momentarily without reversing direction.
Why: Students associate "negative acceleration" with "deceleration" and assume it means slowing down regardless of velocity's sign.
Why: When working quickly, students forget that acceleration is the derivative of velocity, so they stop at the first derivative of position.
Why: Students solve the equation $v(t)=0$ but forget the context that $t \geq 0$ for all motion problems.
Why: Students are used to working with velocity, so they just copy the velocity value (including the negative sign) as the answer for speed.