Why study particle motion calculus? On the AP® exam, free-response questions often ask about the way a particle slides along a track or a car travels on a straight highway. In real-world physics, the same skills predict where satellites, cars, or athletes will be next. This guide explains rectilinear motion, shows how derivatives unlock position, velocity, and acceleration, and offers step-by-step examples that match exam style.

Rectilinear Motion Refresher

Rectilinear motion means movement in a straight line. Three linked functions describe it:

• Position: s(t)
• Velocity: v(t)=s'(t)
• Acceleration: a(t)=v'(t)=s''(t)

Typical units are meters (m) and seconds (s). Therefore, velocity is m/s, while acceleration is m/s².

Example #1

Given s(t)=3t^{2}-12t+5 (0 ≤ t ≤ 6), find v(t) and a(t).

Solution

1. Differentiate once for velocity.
   • v(t)=s'(t)=6t-12
2. Differentiate again for acceleration.
   • a(t)=v'(t)=6

So the particle moves with a constant acceleration of 6 m/s².

Interpreting Position, Velocity, and Acceleration

Signs matter:

• Positive position → right of the origin
• Negative position → left of the origin

Similarly:

• Positive velocity → moving right
• Negative velocity → moving left

For acceleration:

• Positive acceleration speeds velocity upward.
• Negative acceleration pulls velocity downward.

Linking the signs reveals speeding up versus slowing down:

• Negative velocity + positive acceleration → slowing down (moving left, but braking)
• Positive velocity + negative acceleration → slowing down (moving right, but braking)
• Same sign for v and a → speeding up

Example #2

A particle has v(t)=6-2t and constant a(t)=-2. Identify intervals where it speeds up or slows down.

Solution

1. Find when velocity changes sign:
   • 6-2t=0\Rightarrow t=3.
2. Make a sign chart.

Interval	v	a	Result
0 < t < 3	+	–	positive velocity and negative acceleration → slowing down
t > 3	–	–	same sign (both negative) → speeding up

Therefore, the particle slows until 3 s, pauses, then speeds up while moving left.

When Does the Particle Change Direction?

A particle reverses direction where velocity equals zero and the sign of velocity changes. Solve v(t)=0, then test intervals with a sign chart or by graphing v(t).

Example #3

Let s(t)=t^{3}-6t^{2}+9t on 0 ≤ t ≤ 4.

1. Find velocity:
   • v(t)=s'(t)=3t^{2}-12t+9=3(t-1)(t-3).
2. Set v(t)=0: t = 1 s and t = 3 s.
3. Sign chart for v(t):

t-range	test point	v-sign
0–1	0.5	+
1–3	2	–
3–4	3.5	+

Velocity changes sign at 1 s and 3 s, so the particle turns around twice.

Total distance traveled:

• s(0)=0
• s(1)=1-6+9=4
• s(3)=27-54+27=0
• s(4)=64-96+36=4

Distances: |4–0| = 4 m, |0–4| = 4 m, |4–0| = 4 m.Therefore, total distance = 12 m.

Speed vs. Velocity

Speed ignores direction, so \text{speed}=|v(t)|. Many exam errors come from mixing the two. When a problem asks for total distance, integrate the absolute value:

\text{Distance}=\int_{a}^{b}|v(t)|dt

Position, Velocity, and Acceleration Graphs

Graphs make the 4.2 position velocity and acceleration connection visual.

• The slope of a position graph at time t equals v(t).
• The slope of a velocity graph equals a(t).

Therefore, if the velocity graph is rising, acceleration is positive; if it is falling, acceleration is negative.

Calculator-Active Strategies

Graphing calculators shorten work:

• Use nDeriv( to evaluate v(t)=s'(t) or a(t)=v'(t) at single points.
• Use fnInt( for \int |v(t)|dt to find total distance quickly.
• Store s(t) in Y₁, then set a table. Therefore, sign changes become obvious, and tracing reveals turnaround times without algebra.

Quick Reference Chart

Term	Definition / Key Feature
Rectilinear motion	Movement along a straight line
s(t)	Position function (m)
v(t)=s'(t)	Velocity function (m/s)
a(t)=v'(t)=s''(t)	Acceleration function (m/s²)
Speed	v(t), always non-negative
Change of direction	Points where v(t)=0 and velocity sign changes
Displacement	\int_{a}^{b}v(t)dt (signed area)
Total distance	\int_{a}^{b}v(t)dt (absolute area)

Conclusion

Particle motion calculus translates derivatives into real movement. By linking position, velocity, and acceleration, students can spot when a particle speeds up, slows down, or reverses course. Remember the sign rules, practice with graphs, and verify work on a calculator. With these habits, rectilinear motion questions on test day will feel like second nature.

Sharpen Your Skills for AP® Calculus AB-BC

Are you preparing for the AP® Calculus exam? We’ve got you covered! Try our review articles designed to help you confidently tackle real-world math problems. You’ll find everything you need to succeed, from quick tips to detailed strategies. Start exploring now!

• 4.1 Interpreting the Meaning of the Derivative in Context
• 4.3 Rates of Change in Applied Contexts Other Than Motion

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