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 Video by Zeeshan Zamurred Variable Acceleration

Edexcel Mechanics Y1 — Variable Acceleration playlist (Zeeshan Zamurred)Watch the full walkthrough 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 $constant$ acceleration. When acceleration varies with time, you instead have expressions like $s = t^{3} - 2 t$ , 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 $t = 0$ .

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.

1Set

v = 0

3 t^{2} = 4

, so

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

Answer

t = \frac{2}{3} \approx 1.15

Formula recap

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

Quantities as functions of time.

initial value = value at t = 0

Initial conditions.

v = 0 \Rightarrow 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”.

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