Using differentiation to find velocity and acceleration

In a nutshell

When displacement is given as a function of time, velocity and acceleration can be worked out using differentiation.

Equations

DESCRIPTION	EQUATION
Velocity of an object by differentiating displacement.	$v=\dfrac{ds}{dt}$
Acceleration of an object by differentiating velocity.	$a= \dfrac{dv}{dt}$
Acceleration of an object by differentiating displacement.	$a=\dfrac{d^2s}{dt^2}$

Variable definitions

QUANTITY NAME	SYMBOL	UNIT NAME	UNIT
$Displacement$	$s$	$Metre$	$m$
$Velocity$	$v$	$Metres\ per\ second$	$ms^{-1}$
$Acceleration$	$a$	$Metres\ per\ second\ squared$	$ms^{-2}$
$Time$	$t$	$Seconds$	$s$

Velocity

Velocity is the rate of change of displacement, therefore velocity can be worked out by differentiating displacement. When $s$ is expressed as a function of $t$ , the formula for velocity is:

$\boxed{v=\dfrac{ds}{dt}}$

Acceleration

Acceleration is the rate of change of velocity, therefore acceleration can be worked out by differentiating velocity. When $v$ is expressed as a function of $t$ , the formula for acceleration is:

$\boxed{a=\dfrac{dv}{dt}}$

Taking the formula for $v$ , the formula for acceleration is also:

$\boxed{a=\dfrac{d^2s}{dt^2}}$

This is the second derivative of displacement as a function of time.

Example

A bullet is shot in a straight line. the displacement of the bullet at time $t$ is modelled by $s=2t^4+3t^2-18t+24$ . Find:

a: The velocity of the bullet when $t= 4$ .

b: The acceleration of the bullet when $t=2$ .

a: Find a formula for velocity by differentiating the formula for displacement:

$s=2t^4+3t^2-18t+24$

$v=\dfrac{ds}{dt} = 8t^3+6t-18$

Substitute $t=4$ :

$v = 8(4^3)+6(4)-18$

$v = 512 + 24 -18 = 518$

The velocity of the bullet at $4s$ is $\underline{518\ ms^{-1}}$ .

b: Find a formula for acceleration by differentiating velocity:

$v=8t^3+6t-18$

$a=\dfrac{dv}{dt} = 24t^2+6$ 

Substitute $t=2$ :

$a = 24(2^2)+6$

$a= 96+6=102$ 

The acceleration of the bullet at $2s$ is $\underline{102\ ms^{-2}}$ .