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Statement of the equations

The "SUVAT" equations refer to the five equations: $$v=u+at$$ $$s=ut+\frac{1}{2}at^2$$ $$v^2=u^2+2as$$ $$s=\frac{1}{2}\left(v+u\right)t$$ $$s=vt-\frac{1}{2}at^2$$ of which the first three are the most commonly taught and used.

These equations are applicable only for a particle moving under the influence of a constant acceleration. Speeds should be non-relativistic.

The name "SUVAT" comes from the variables used:

  • s: Displacement of particle (in time t)
  • u: Initial velocity
  • v: Final velocity (after time t)
  • a: Acceleration
  • t: Time

The equations hold for vectors as well.

Derivation of the equations

The first equation can be obtained by a simple rearrangement of the definition of constant acceleration:

$$a = \frac{v-u}{t} \implies v = u + at$$

The second equation results from the integration of the above one with respect to $t$:

$$\int v dt = \int u dt + \int a t dt \implies s = ut + \frac{1}{2}at^2$$

The third equation is a result of eliminating the variable $t$ between the first two equations.

$$t=\frac{v-u}{a} \implies s = \frac{u(v-u)}{a} + \frac{a(v-u)^2}{2a^2} \implies v^2 = u^2 + 2as$$

The fourth equation is a result of eliminating the variable $a$ between the first two equations.

$$a=\frac{v-u}{t} \implies s = ut + \frac{(v-u)t^2}{2t}$$

The final equation is a result of eliminating the variable $u$ between the first two equations.

$$u = v-at \implies s = (v-at)t + \frac{1}{2}at^2 \implies s = vt - \frac{1}{2}at^2$$

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