Greatest Displacement and Greatest Speed

On a closed time interval, a greatest or least value may occur at an endpoint. For displacement, compare \(s\) at the endpoints and every admissible stationary time \(v=0\). For greatest distance from the origin, compare \(|s|\) instead; for distance from the starting point, compare \(|s-s(0)|\). These are different quantities.

For maximum speed, compare \(|v|\) at the endpoints and at interior candidates where \(a=0\), also checking any joins or points where the formula is not differentiable. Rest times \(v=0\) are possible minimum-speed points. The greatest velocity need not give the greatest speed.

For illustration, let \(v=2-t\) for \(0\le t\le5\). The greatest velocity is \(2\) at the start, but the greatest speed is \(3\) at the end. The particle rests at \(t=2\), where its direction changes.

To decide whether speed is increasing, examine the signs of velocity and acceleration: equal nonzero signs mean increasing speed; opposite signs mean decreasing speed. Negative acceleration can increase speed when velocity is negative. At a zero or join, examine the neighbouring intervals rather than relying on one product value.