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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 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

  • 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
1

Deriving v = u + at

If acceleration aa is constant, integrating it with respect to time gives velocity. The constant of integration is the initial velocity uu.

v=∫a dt=at+uv = \int a \, dt = at + u
Integrating constant aa reproduces v=u+atv = u + at.
2

Deriving s = ut + ½at²

Integrating that velocity expression gives displacement, with the constant being the initial displacement (taken as 00 here).

s=∫(u+at) dt=ut+12at2s = \int (u + at)\, dt = ut + \tfrac{1}{2}at^{2}
Integrating v=u+atv = u + at gives the SUVAT displacement formula.
Example 1Show integrating v=u+atv = u + at gives s=ut+12at2s = ut + \tfrac{1}{2}at^2.
1∫u dt=ut\int u \, dt = ut; ∫at dt=12at2\int at \, dt = \tfrac{1}{2}at^2.
2Constant is s0=0s_0 = 0.
Answers=ut+12at2s = ut + \tfrac{1}{2}at^{2} ∎

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

3

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=∫a dt=u+atv = \int a \, dt = u + at
Derive v = u + at.
s=∫(u+at) dt=ut+12at2s = \int (u + at)\,dt = ut + \tfrac{1}{2}at^{2}
Derive s = ut + ½at².
constant a⇒SUVAT=calculus\text{constant } a \Rightarrow \text{SUVAT} = \text{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).

Test yourself

Answer a few questions, then review anything you missed.

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Keep practising

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