M11.1MechanicsStretch

Functions of Time

When acceleration is not constant, SUVAT no longer works. Instead, displacement, velocity and acceleration are written as functions of time, and calculus links them — the bridge between Pure and Mechanics.

20 min Variable Acceleration

Edexcel Mechanics Y1 — Variable Acceleration playlist (Zeeshan Zamurred)Curated by Kepler, not produced by us. Watch it before the notes below.

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What you'll be able to do

Express s, v and a as functions of time
Evaluate these at a given time
Understand why SUVAT fails here
Find values like the initial velocity

Why not SUVAT?

SUVAT equations assume $\textbf{constant}$ acceleration. When acceleration varies with time, you instead have expressions like $s = t^3 - 2t$ , and you use calculus to move between displacement, velocity and acceleration.

Evaluating at a time

Given a function of time, substitute the value of $t$ to find the quantity at that instant. For example, the initial value is found at $t = 0$ .

Example 1A particle has

v = 3t^2 - 4

(m/s). Find the velocity at

t = 2

1Substitute

t = 2

3(4) - 4

Answer

8

m/s

Tip — The “initial” value of any quantity is just its value at t = 0.

Setting up problems

Often you are given one function and asked for things like “when is the particle at rest?” (set $v = 0$ ) or “the initial acceleration” (the acceleration function at $t = 0$ ). Translating words into a time-substitution or equation is the key skill.

Example 1For

v = 3t^2 - 4

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

1Set

v = 0

3t^2 = 4

, so

t^2 = \tfrac{4}{3}

Answer

t = \tfrac{2}{\sqrt{3}} \approx 1.15

Formula recap

s = f(t), \ v = g(t), \ a = h(t)

Quantities as functions of time.

\text{initial value} = \text{value at } t = 0

Initial conditions.

v = 0 \Rightarrow \text{at rest}

Instantaneously at rest.

Common mistakes to avoid

Using SUVAT when acceleration varies.

SUVAT needs constant acceleration; use calculus instead.

Confusing “at rest” (v = 0) with “at the origin” (s = 0).

At rest means velocity 0; at the origin means displacement 0.

Key takeaways

Variable acceleration ⟶ s, v, a are functions of time (no SUVAT).
Substitute t to evaluate; t = 0 gives initial values.
Set v = 0 for “at rest”; s = 0 for “at the origin”.

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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