Problems

Solution: Considering the horizontal component, this will be constant as there are no forces acting in that direction. The first step will take the particle \(t = \sqrt{\frac{2h}g}\) to reach. At which point it will be travelling with speed \(v = \sqrt{2gh} \) (by energy considerations, \(mgh = \frac12 mv^2\)). To reach the second step must take twice as long (since the ball has to travel \(2d\) horizontally, rather than \(d\)). Since \(t = 2\sqrt{\frac{2h}g}\) we must have that: \begin{align*} && s &= ut + \frac12 gt^2 \\ \Rightarrow && h &= u 2\sqrt{\frac{2h}g} + \frac12 g \frac{8h}g \\ \Rightarrow && u &= -\frac{3}{2} h\sqrt{\frac{g}{2h}} \\ &&&= -\frac{3}{2\sqrt{2}} \sqrt{gh} \end{align*} Therefore, using Newton's experimental law, we must have that \(e = \frac{\frac{3}{2 \sqrt{2}} \sqrt{gh}}{\sqrt{2} \sqrt{gh}} = \frac{3}{4}\). Again by conservation of energy \(mgh + \frac12 \frac{9}{8} mgh = \frac12 mv^2 \Rightarrow v = \frac{5}{2\sqrt{2}} \sqrt{gh}\) when it lands on the next step. Therefore we would need the coefficient of restitution for the second (and subsequent steps) to be: \(\displaystyle \frac{\frac{3}{2\sqrt{2}} \sqrt{gh}}{\frac{5}{2\sqrt{2}} \sqrt{gh}} = \frac35\)