Straight-Line Motion: Position, Velocity, Acceleration
Analyze the motion of a particle on a line. Determine when the particle is at rest, moving in each direction, speeding up, or slowing down using the signs of v(t) and a(t). Compute total distance and displacement.
A lesson of AP Calculus AB, in Contextual Applications of Differentiation, about 10 minutes.
Motion Analysis
By the end of this lesson, you will be able to analyze the motion of a particle using position, velocity, and acceleration functions, and determine when a particle is moving left/right, speeding up/slowing down, and at rest.
Position, Velocity, and Acceleration
If $s(t)$ is the position of a particle at time t:
Velocity: $v(t) = s'(t)$ — the rate of change of position; positive = moving right, negative = moving left.
Acceleration: $a(t) = v'(t) = s''(t)$ — the rate of change of velocity; positive = accelerating in the positive direction.
Speed: $|v(t)|$ — always non-negative; the magnitude of velocity without direction.
Key Interpretations
| Condition | Meaning |
|---|---|
| $v(t) > 0$ | Moving in positive direction (right) |
| $v(t) < 0$ | Moving in negative direction (left) |
| $v(t) = 0$ | Momentarily at rest |
| $v(t)$ and $a(t)$ same sign | Speeding up |
| $v(t)$ and $a(t)$ opposite sign | Slowing down |
| $a(t) = 0$, $v(t) \neq 0$ | Constant velocity (uniform motion) |
Total Distance vs. Displacement
Displacement (net change in position) = $s(b) - s(a) = \int_a^b v(t), dt$
Total distance traveled = $\int_a^b |v(t)|, dt$
These are different whenever the particle changes direction (v changes sign). A particle that goes 5 right then 3 left has displacement +2 but traveled a total distance of 8.
Example
A particle moves along the x-axis with position $s(t) = t^3 - 6t^2 + 9t + 2$ for $t \geq 0$.
Find v(t) and a(t):
$v(t) = 3t^2 - 12t + 9 = 3(t^2 - 4t + 3) = 3(t-1)(t-3)$
$a(t) = 6t - 12 = 6(t-2)$
When is the particle at rest? $v(t) = 0$ at $t = 1$ and $t = 3$.
Direction of motion:
- $0 < t < 1$: $v > 0$ (moving right)
- $1 < t < 3$: $v < 0$ (moving left)