Deriving the suvat Formulae

Velocity-Time Gradient: For an object with constant acceleration, the velocity-time graph is a straight line. The gradient of this line represents the constant acceleration: $a = \frac{v - u}{t}$.

Area Under the Graph: The displacement ($s$) of an object is represented by the area bounded by the velocity-time line and the time axis. In the case of constant acceleration, this area forms a trapezium.

Base Formulae: The two primary starting points for all derivations are the definition of acceleration ($v = u + at$) and the area of a trapezium for displacement ($s = \frac{1}{2}(u + v)t$).

Eliminating Time ($t$): To derive $v^2 = u^2 + 2as$, start with $t = \frac{v - u}{a}$ and substitute it into the displacement formula $s = \frac{u + v}{2}t$. This results in $s = \frac{(u + v)(v - u)}{2a}$, which simplifies using the difference of two squares.

Eliminating Final Velocity ($v$): By substituting $v = u + at$ into $s = \frac{1}{2}(u + v)t$, we get $s = \frac{1}{2}(u + u + at)t$, which simplifies to the widely used $s = ut + \frac{1}{2}at^2$.

Eliminating Initial Velocity ($u$): Conversely, substituting $u = v - at$ into the area formula yields $s = vt - \frac{1}{2}at^2$, which is useful when the initial speed is unknown.

The 'List' Method: Always begin a problem by writing the letters s, u, v, a, t and filling in the three known values. This identifies the one variable you need to find and the one variable that is irrelevant for that specific step.

Unit Consistency: Ensure all values are in consistent SI units (meters, seconds, $m/s$, $m/s^2$) before substituting into equations to avoid magnitude errors.

Sanity Checking: If an object is 'decelerating', your calculated value for $a$ should be opposite in sign to the velocity. If an object 'comes to rest', set $v = 0$ immediately.

Variable Acceleration: A frequent mistake is attempting to use suvat formulae for motion where acceleration changes. In such cases, calculus (integration/differentiation) must be used instead.

Misinterpreting 'at rest': Students often forget that 'initially at rest' implies $u = 0$, or 'returns to start' implies $s = 0$, missing vital pieces of information hidden in the text.

Squared Term Errors: In the formula $v^2 = u^2 + 2as$, students often forget that $u^2$ is a separate term and incorrectly apply the square to $(u + 2as)$.