SUVAT Equations, also known as the equations of motion, relate the variables of displacement(s), initial velocity(u), final velocity(v), acceleration(a), and time(t). There are three such equations.

Equation 1

Equation 1 describes the relation between u, a, and t with v.

$$ v = u+at $$

Derivation

$$ a = \frac{\Delta v}{t}\newline a = \frac{v-u}{t}\text{ \{expanding change in velocity\}}\newline at = v - u\newline v = u + at $$

Equation 2

Equation 2 describes the relation between u, t, and a with s.

$$ s = ut+ \frac{at^2}2 $$

Derivation

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$$ s = \text{Area under the graph} = A+B\newline s = (ut)+[(v-u)t\times\frac12]\newline s=ut[(at)t\times\frac12]\text{ \{}v-u=at\text{, from equation 1\}}\newline s=ut+\frac{at^2}2 $$

Equation 3

Equation 3 describes the relation between a and s with v and u.

$$ 2as=v^2-u^2 $$

Derivation

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$$ s = \text{Area under the graph} = A+B\newline s = \frac{v+u}2t\newline s=\frac{v+u}2\times\frac{v+u}a \text{ \{}t=\frac{v+u}a\text{, from velocity equation\}}\newline 2as=v^2-u^2 $$