4.2 Straight-Line Motion

Position, velocity, acceleration

For an object on a line: velocity v(t) = s'(t) (derivative of position), and acceleration a(t) = v'(t) = s''(t). Speed = |v(t)|, the absolute value of velocity.

Direction & speeding up

The object moves in the positive direction when v > 0, negative when v < 0, and is momentarily at rest when v = 0. It is speeding up when v and a have the same sign, and slowing down when they have opposite signs.

Key terms

Velocity

v(t) = s'(t), the derivative of position.

Acceleration

a(t) = v'(t) = s''(t).

Speed

|v(t)| — the magnitude of velocity.

Speeding up

v and a have the same sign.

At rest

Velocity v(t) = 0.

Try an AP-style question

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

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