Average and instantaneous velocity

Warm-up

Three quick questions from earlier lessons.

Discover

A ball is thrown up. Its height after $t$ seconds is $s(t)$ meters. Here is the line through the points of the graph at $t = 1$ and $t = 1 + h$ (the curve is in violet). Make $h$ smaller.

The height s(t) = 20t − 5t² and the line through its points at t = 1 and t = 1 + h. As h shrinks, the line settles on the tangent at t = 1, of slope 10.

The slope of that line is the average velocity on $[1, 1 + h]$ . As $h$ shrinks, what does it approach?

About 10 m/s
15

Key idea

Velocity at one instant is the limit of average velocities over shorter and shorter intervals.

Average velocity on

[a, a + h]

:

\dfrac{s(a + h) - s(a)}{h}

. Instantaneous velocity at

t = a

:

\displaystyle\lim_{h \to 0} \dfrac{s(a + h) - s(a)}{h}

.

Worked example

$s(t) = 20t - 5t^2$ . Find the velocity at $t = 1$ .

s(1 + h) = 15 + 10h - 5h^2

Substitute 1 + h and expand: 20 + 20h − 5(1 + 2h + h²).

\dfrac{s(1 + h) - s(1)}{h} = 10 - 5h

Subtract s(1) = 15 and divide by h ≠ 0.

velocity at

t = 1

:

10

m/s

Let h → 0.

Your turn, guided

Your turn: same ball, $t = 3$ . The average velocity on $[3, 3 + h]$ simplifies to $-10 - 5h$ .

velocity at

t = 3

:

-10

m/s

Let h → 0: the ball is on its way down.

Practice ★

A ball's position is $s(t) = 4t^2 + 9t$ meters after $t$ seconds. Find its average velocity on $[0, 6]$ , in m/s.
A ball's position is $s(t) = 1t^2 + 4t$ meters after $t$ seconds. Find its average velocity on $[0, 5]$ , in m/s.
A ball's position is $s(t) = 3t^2 + 9t$ meters after $t$ seconds. Find its average velocity on $[0, 4]$ , in m/s.

Watch out!

Practice ★★

$s(t) = 5t^2$ . Simplify the average velocity on $[3, 3 + h]$ , $\dfrac{s(3 + h) - s(3)}{h}$ for $h \ne 0$ . Enter an expression in $h$ .
$s(t) = 2t^2$ . Simplify the average velocity on $[2, 2 + h]$ , $\dfrac{s(2 + h) - s(2)}{h}$ for $h \ne 0$ . Enter an expression in $h$ .

$s(t) = 3t^2 + 7t$ meters. The average velocity on $[6, 6 + h]$ simplifies to $43 + 3h$ m/s. Find the instantaneous velocity at $t = 6$ , in m/s.
$s(t) = 1t^2 + 6t$ meters. The average velocity on $[6, 6 + h]$ simplifies to $18 + 1h$ m/s. Find the instantaneous velocity at $t = 6$ , in m/s.

Exit ticket

Three questions to check you have got it.

$s(t) = 3t^2 + 7t$ meters. The average velocity on $[6, 6 + h]$ simplifies to $43 + 3h$ m/s. Find the instantaneous velocity at $t = 6$ , in m/s.
$s(t) = 1t^2 + 6t$ meters. The average velocity on $[6, 6 + h]$ simplifies to $18 + 1h$ m/s. Find the instantaneous velocity at $t = 6$ , in m/s.
$s(t) = 5t^2 + 5t$ meters. The average velocity on $[5, 5 + h]$ simplifies to $55 + 5h$ m/s. Find the instantaneous velocity at $t = 5$ , in m/s.

Summary card

Velocity

Average: $\dfrac{s(a + h) - s(a)}{h}$ , change in position ÷ change in time.
Instantaneous: the limit of that as $h \to 0$ .

Watch out: divide by the time; a change in position alone is not a velocity.

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