Deriving the suvat Formulas

On a velocity-time (v-t) graph, the gradient of the line represents acceleration. For constant acceleration, the graph is a straight line with equation $a = \frac{v - u}{t}$, which rearranges to the first formula: $v = u + at$.

The area under the v-t graph represents the displacement ($s$). This area forms a trapezium with parallel sides of lengths $u$ and $v$, and a height of $t$.

Using the area of a trapezium formula, we derive: $s = \frac{1}{2}(u + v)t$.

By splitting the area into a rectangle (base $t$, height $u$) and a triangle (base $t$, height $v-u$), we derive: $s = ut + \frac{1}{2}(v - u)t$. Substituting $(v - u) = at$ yields $s = ut + \frac{1}{2}at^2$.

Acceleration is defined as the rate of change of velocity: $a = \frac{dv}{dt}$. Integrating both sides with respect to time gives $v = \int a \, dt = at + C$.

Applying the initial condition that at $t = 0$, velocity is $u$, we find the constant $C = u$, resulting in $v = u + at$.

Velocity is the rate of change of displacement: $v = \frac{ds}{dt}$. Integrating the velocity function gives $s = \int (u + at) \, dt = ut + \frac{1}{2}at^2 + C$.

Assuming the particle starts at the origin ($s = 0$ at $t = 0$), the constant $C$ becomes zero, yielding $s = ut + \frac{1}{2}at^2$.

The formula $v^2 = u^2 + 2as$ is unique because it does not involve the time variable ($t$).

To derive it, we start with $v = u + at$ and rearrange for time: $t = \frac{v - u}{a}$.

This expression for $t$ is then substituted into the displacement formula $s = \frac{1}{2}(u + v)t$.

Multiplying the terms results in $s = \frac{(u + v)(v - u)}{2a} = \frac{v^2 - u^2}{2a}$, which rearranges to $v^2 = u^2 + 2as$.

Identify the 'Missing' Variable: Each of the five equations lacks exactly one of the five suvat variables. Choose the equation that excludes the variable you neither know nor need to find.

Define a Positive Direction: Always explicitly state which direction is positive (e.g., 'upwards is positive'). This ensures that vectors like gravity ($g$) are assigned the correct sign ($-9.8$ if upwards is positive).

Check Units: Ensure all quantities are in standard SI units (meters, seconds, $ms^{-1}$, $ms^{-2}$) before substituting into formulas.

Sanity Check: If an object is decelerating, ensure your calculated final velocity is lower than the initial velocity, or that the acceleration value is negative relative to the velocity.