A particle strikes a horizontal smooth floor with a velocity u making an angle $\theta$ with the floor and rebounds with velocity v making an angle $\varphi$ with the floor. If the coefficient of restitution between the particle and the floor is e, then

$$\begin{gathered} \mathrm{vsin}\mathrm{\phi }=\mathrm{eusin}\mathrm{\theta }---\left(1\right),\mathrm{vcos}\mathrm{\phi }=\mathrm{ucos}\mathrm{\theta }----\left(2\right) \\ \left(1\right)/\left(2\right) \\ \frac{\mathrm{vsin}\mathrm{\phi }}{\mathrm{vcos}\mathrm{\phi }}=\frac{\mathrm{eusin}\mathrm{\theta }}{\mathrm{ucos}\mathrm{\theta }} \\ \mathrm{tan\phi }=\mathrm{etan\theta } \end{gathered}$$

$\begin{array}{l}\mathrm{squaring} \mathrm{and} \mathrm{adding} \left(1\right) \& \left(2\right) \mathrm{we} \mathrm{get} \\ \mathrm{v}=\mathrm{u}\sqrt{{\cos }^2\mathrm{\theta }+{\mathrm{e}}^2{\sin }^2\mathrm{\theta }}=\mathrm{u}\sqrt{1-{\sin }^2\mathrm{\theta }+{\mathrm{e}}^2{\sin }^2\mathrm{\theta }}\\ =\mathrm{u}\sqrt{1-\left(1-{\mathrm{e}}^2\right){\sin }^2\mathrm{\theta }}\\ \mathrm{I}=\mathrm{m}(\mathrm{vsin}\mathrm{\varphi }+\mathrm{usin}\mathrm{\theta })=\mathrm{musin}\mathrm{\theta }(1+\mathrm{e})\end{array}$

Ratio of KE $=\frac{\frac{1}{2}{\mathrm{mv}}^2}{\frac{1}{2}{\mathrm{mu}}^2}={\cos }^2\mathrm{\theta }+{\mathrm{e}}^2{\sin }^2\mathrm{\theta }$