Unit 4.2 – Straight-Line Motion: Connecting Position, Velocity, and Acceleration
AP® Calculus AB & BC | Real-World Motion and Rates of Change
Understanding the calculus of motion is central to science and AP® Calculus! In straight-line motion, a particle's position, velocity, and acceleration are all connected through differentiation.
🚗 Fundamental Definitions & Formulas
Position Function
Let \( s(t) \) = position at time \( t \) (units: e.g., meters).
First Derivative: Velocity
\[ v(t) = s'(t) = \frac{ds}{dt} \]
Units: meters/second (or whatever position/time units are used).
Meaning: The rate of change of position—i.e., how fast and in what direction the object moves.
Meaning: The rate of change of position—i.e., how fast and in what direction the object moves.
Second Derivative: Acceleration
\[ a(t) = v'(t) = s''(t) = \frac{dv}{dt} = \frac{d^2s}{dt^2} \]
Units: meters/second2.
Meaning: The rate of change of velocity—how quickly the velocity is increasing or decreasing.
Meaning: The rate of change of velocity—how quickly the velocity is increasing or decreasing.
🔗 Calculus Connections in Motion
- Position → derivative ⇒ Velocity → derivative ⇒ Acceleration
- \( v(t) > 0 \): moving right/forward; \( v(t) < 0 \): moving left/backward
- \( a(t) > 0 \): velocity increasing; \( a(t) < 0 \): velocity decreasing
- Speed = \( |v(t)| \): always non-negative
- When \( v(t) = 0 \): particle is stopped/at rest
📖 Common AP® Contexts
When is a particle speeding up or slowing down?
- Speeding up: Velocity and acceleration have the same sign.
- Slowing down: Velocity and acceleration have opposite signs.
Finding total distance traveled:
Integrate or sum \( |v(t)| \) over time intervals. Always find where \( v(t) \) changes sign first!
Instantaneous velocity and acceleration at a point:
Plug \( t=a \) into \( v(t) \) for velocity, and into \( a(t) \) for acceleration.
Maximum/Minimum Position or Speed:
Set \( v(t)=0 \) to find critical points. Test endpoints and critical points for position (\(s(t)\)) or speed (\(|v(t)|\)).
🎯 Motion Short Notes & Tricks
- Velocity is signed (it indicates direction); speed is always non-negative.
- Use sign charts for \( v(t) \) and \( a(t) \) to analyze motion phases.
- "Speeding up" occurs when \( v(t) \) and \( a(t) \) are both positive or both negative.
- For distance vs. displacement:
- Displacement = \( s(b) - s(a) \)
- Distance Traveled = Sum of all absolute changes in position (account for turning points where \(v(t)=0\))!
- Critical numbers for the position function occur when \( v(t)=0 \) or at endpoints of the interval.
- Write units everywhere! AP® points depend on it.
🧮 Worked Examples
Example 1: Position to Velocity & Acceleration
If \( s(t) = t^3 - 6t^2 + 9t + 2 \), then:
\[ v(t) = 3t^2 - 12t + 9 \]
\[ a(t) = 6t - 12 \]
Example 2: Interpreting Signs
If \( v(4) < 0 \) and \( a(4) > 0 \), the signs are opposite, so the particle is moving left but slowing down. If both were negative, it would be moving left and speeding up.
Example 3: Finding Total Distance
Given velocity \( v(t) = 4 - 2t \) on the interval \( 0 \leq t \leq 5 \).
The velocity changes sign when \( v(t) = 0 \), which is at \( t=2 \).
The velocity changes sign when \( v(t) = 0 \), which is at \( t=2 \).
To compute total distance, sum the absolute displacements on the sub-intervals:
\[ \text{Distance} = |s(2) - s(0)| + |s(5) - s(2)| \]Alternatively, integrate the speed, \(|v(t)|\), over the entire interval.
📝 Practice Problems
Try These Yourself:
- If \( s(t) = 2t^3 - 9t^2 + 12t \), find \( v(t) \) and \( a(t) \).
- Given \( v(t)=t^2-4 \), when is the object at rest? When is it moving right? Moving left?
- Let \( s(t)=\sin(t) \) (position in meters, \(t\) in seconds). What are the units of, and formula for, \(a(t)\)?
Solutions:
- \( v(t)=6t^2-18t+12 \), and \( a(t)=12t-18 \).
- The object is at rest when \( v(t)=0 \), so at \(t=2\) (assuming \(t \geq 0\)). It moves right when \(v(t)>0\) (for \(t>2\)) and left when \(v(t)<0\) (for \(0 \leq t < 2\)).
- The velocity is \(v(t)=\cos(t)\), and acceleration is \(a(t)=-\sin(t)\). The units of acceleration are meters/second\(^2\).
✏️ AP® Exam Tips for Motion Problems
- Always specify both direction (if applicable) and units in your answers!
- Include a sign chart when analyzing particle motion over intervals.
- For total distance, explicitly find when \(v(t)=0\) and sum the absolute displacements.
- Box or highlight final answers for stopping time, distance, or speed for clarity in FRQ responses.
- Justify your conclusions using calculus (e.g., "The particle is slowing down because \(v(t)\) and \(a(t)\) have opposite signs...").
- Clearly label \(t\) values in all your work, including endpoints and critical numbers.