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

Variable definitions

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 $$\underline{102\ ms^{-2}}$$.​