4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration

Position, Velocity, Acceleration

For an object moving along a line, position s(t) gives location, velocity v(t) = s'(t) gives rate of position change, and acceleration a(t) = v'(t) = s''(t) gives rate of velocity change. Velocity's sign shows direction: positive means moving in the positive direction, negative means moving in the negative direction.

Speed and Speeding Up/Slowing Down

Speed = |v(t)|, the magnitude of velocity. An object is speeding up when velocity and acceleration have the same sign (both positive or both negative), and slowing down when they have opposite signs. This is because acceleration in the same direction as motion increases speed.

Key Motion Moments

The object is at rest or changes direction when v(t) = 0 (and v changes sign around that point). Total distance traveled requires integrating |v(t)|, while displacement is the integral of v(t) itself. Checking v(t) and a(t) together tells the complete motion story at any instant.

Key terms

Velocity v(t)

The derivative of position, s'(t); indicates speed and direction of motion.

Acceleration a(t)

The derivative of velocity, v'(t), or second derivative of position, s''(t).

Speed

The absolute value of velocity, |v(t)|; always non-negative.

Speeding up condition

Velocity and acceleration have the same sign.

Slowing down condition

Velocity and acceleration have opposite signs.

Try an AP-style question

A particle moves with s(t) = t³ − 6t² + 9t. When is it at rest?

1. t = 1 and t = 3
2. t = 2 only
3. t = 0 and t = 3
4. t = 0, 3, and 9