M11.2MechanicsStretch

Using Differentiation

Differentiation steps you down the kinematics chain: differentiate displacement to get velocity, and velocity to get acceleration. Each derivative is a rate of change with respect to time.

25 min Variable Acceleration

Edexcel AS Level Maths: 11.2 Using DifferentiationCurated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Differentiate displacement to find velocity
Differentiate velocity to find acceleration
Find velocity and acceleration at a given time
Find when the particle is at rest

The differentiation chain

Velocity is the rate of change of displacement, and acceleration is the rate of change of velocity. So differentiate once to go from $s$ to $v$ , and again to go from $v$ to $a$ .

v = \dfrac{ds}{dt}, \qquad a = \dfrac{dv}{dt} = \dfrac{d^2s}{dt^2}

Differentiate down:

s \to v \to a

Example 1If

s = t^3 - 2t^2

(m), find

v

and

a

v = \dfrac{ds}{dt} = 3t^2 - 4t

a = \dfrac{dv}{dt} = 6t - 4

Answer

v = 3t^2 - 4t

a = 6t - 4

Evaluating at a time

After differentiating, substitute the time to get the velocity or acceleration at that instant.

Tip — Differentiate FIRST to get the v or a function, THEN substitute the time value.

When is it at rest?

The particle is instantaneously at rest when $v = 0$ . Differentiate $s$ to get $v$ , set it to zero, and solve for $t$ .

Example 1For

s = t^3 - 2t^2

, when is the particle at rest? (t ≥ 0)

v = 3t^2 - 4t = t(3t - 4) = 0

t = 0

t = \tfrac{4}{3}

Answer

t = 0

and

t = \tfrac{4}{3}

Formula recap

v = \dfrac{ds}{dt}

Velocity = derivative of displacement.

a = \dfrac{dv}{dt} = \dfrac{d^2 s}{dt^2}

Acceleration = derivative of velocity.

v = 0 \Rightarrow \text{at rest}

Find rest by setting v = 0.

Common mistakes to avoid

Differentiating to go from velocity to displacement.

Differentiation goes s → v → a; integration goes the other way.

Substituting the time before differentiating.

Differentiate to get the function first, then substitute.

Key takeaways

v = ds/dt and a = dv/dt = d²s/dt².
Differentiate to step down: displacement → velocity → acceleration.
At rest ⟺ v = 0.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

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Related lessons

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Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.