Motion on a line: velocity, acceleration, distance

Warm-up

Three quick questions from earlier lessons.

Discover

A ball moving left has velocity $-4$ m/s; its acceleration is $-2$ m/s². Is it getting faster or slower?

Faster
Slower, since the acceleration is negative

Key idea

For a particle on a line with position $s(t)$ :

v(t) = s'(t)

,

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

. At rest:

v = 0

. It changes direction where

v

changes sign. Speeding up when

v

and

a

have the same sign, slowing down when opposite. Total distance adds the distances of each leg between turns.

Worked example

$s(t) = t^2 - 4t$ on $[0, 5]$ . Find the total distance.

It turns at

t = 2

v = 2t − 4 = 0.

Legs:

|-4 - 0| = 4

and

|5 - (-4)| = 9

s(0) = 0, s(2) = −4, s(5) = 5.

Total distance

13

m

Add the legs. (The displacement is only 5.)

Your turn, guided

Your turn: $s(t) = t^2 - 6t$ . When is it at rest?

t = 3

v(t) = 2t − 6 = 0.

Practice ★

A particle's position is $s(t) = t^2 - 2t$ . At what time $t$ is it at rest?
A particle's position is $s(t) = t^2 - 6t$ . At what time $t$ is it at rest?

At $t = 3$ , a particle has velocity $-8$ m/s and acceleration $-2$ m/s². Is it speeding up or slowing down?
1. Speeding up: velocity and acceleration have the same sign
2. Slowing down, because the acceleration is negative
3. Neither: its speed stays the same

Watch out!

Practice ★★

A particle's position is $s(t) = t^2 - 4t$ meters. Find the total distance it travels from $t = 0$ to $t = 9$ .
A particle's position is $s(t) = t^2 - 4t$ meters. Find the total distance it travels from $t = 0$ to $t = 6$ .

Exit ticket

Three questions to check you have got it.

A particle's position is $s(t) = t^2 - 2t$ . At what time $t$ is it at rest?
A particle's position is $s(t) = t^2 - 6t$ . At what time $t$ is it at rest?
A particle's position is $s(t) = t^2 - 10t$ . At what time $t$ is it at rest?

Summary card

Motion on a line

$v = s'$ , $a = v'$ ; at rest when $v = 0$ .
Same signs of $v$ and $a$ : speeding up.
Distance: add each leg between turns.

Watch out: displacement and distance differ when the particle turns back.

About this lesson