M11.5MechanicsStretch

Constant Acceleration Formulae

The calculus of this chapter ties back to the SUVAT formulas: if acceleration is constant, integrating it reproduces v = u + at and s = ut + ½at². This shows the SUVAT equations are a special case of the calculus methods.

25 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

See SUVAT as a special case of calculus
Derive v = u + at by integrating constant a
Derive s = ut + ½at² by integrating v
Connect calculus and SUVAT methods

Deriving v = u + at

If acceleration $a$ is constant, integrating it with respect to time gives velocity. The constant of integration is the initial velocity $u$ .

v = \int a d t = a t + u

Integrating constant

a

reproduces

v = u + a t

Deriving s = ut + ½at²

Integrating that velocity expression gives displacement, with the constant being the initial displacement (taken as $0$ here).

s = \int (u + a t) d t = u t + \frac{1}{2} a t^{2}

Integrating

v = u + a t

gives the SUVAT displacement formula.

\int u d t = u t

;

\int a t d t = \frac{1}{2} a t^{2}

2Constant is

s_{0} = 0

Answer

s = u t + \frac{1}{2} a t^{2}

∎

Tip — SUVAT is just calculus with a constant a — useful insight for “show that” questions.

Choosing the right method

If acceleration is constant, either SUVAT or calculus works (SUVAT is quicker). If acceleration varies with time, you must use calculus. Recognising which situation you are in is the key decision.

Formula recap

v = \int a d t = u + a t

Derive v = u + at.

s = \int (u + a t) d t = u t + \frac{1}{2} a t^{2}

Derive s = ut + ½at².

constant a \Rightarrow SUVAT = calculus

Special case.

Common mistakes to avoid

Using SUVAT for a time-varying acceleration.

SUVAT only applies when a is constant; otherwise integrate.

Dropping the initial-velocity constant when integrating a.

The constant of integration is u (the initial velocity).

Key takeaways

Integrating a constant acceleration reproduces v = u + at.
Integrating that velocity reproduces s = ut + ½at².
SUVAT is a special case of the calculus methods (constant a).

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