Kinematics - Constant-Acceleration Motion

In physics, this is often associated with the concept of critical points in the velocity-time graph, where the slope (which represents acceleration) is zero. A peak or trough is an extremum of signed velocity. It need not be the corresponding extremum of speed, which is the magnitude of velocity.

For straight-line motion with constant acceleration \(a\), let \(u\) be initial velocity, \(v\) velocity after time \(t\), and \(s\) the net displacement from the starting point. The four equations are

1. \(v = u + at\) or \(v_{f} = v_{i} + at\)
2. \(s = ut + \frac{1}{2}at^{2}\)
3. \(v^{2} = u^{2} + 2as\)
4. \(s = \left( \frac{u + v}{2} \right)t\)

These familiar equations allow students to analyse and predict the motion of objects without having to perform tedious calculus.
Prove these four equations using calculus.

Note: We will use these equations in later questions.