Question

A small smooth ball is falling vertically downward as shown in figure. The ball strikes a smooth plane which is inclined at an angle 30 to the horizontal. Immediately before striking the plane the ball has speed of 8 $m{s}^{-1}$. The coefficient of restitution 1 between the ball and plane is $\frac{1}{4}$. The speed of the ball just after the impact is:

Answer 1

Elastic collision in $2D$ Before the impact:-

After the impact:

$\sin {30}^o=\frac{1}{2}$

$\sin \alpha =\frac{1}{2}$

$\cos \alpha =\frac{\sqrt{3}}{2}$

$\tan \alpha =\frac{1}{\sqrt{3}}$

Component of vel $l{l}^{le}$ to the slope is $8\sin \alpha \Rightarrow 8\times \frac{1}{2}=4$ ${\perp }^{lar}$ to slope $\to v=e\times 8\cos \alpha$

$\frac{1}{4}\times 8\times \frac{\sqrt{3}}{2}\Rightarrow \sqrt{3}$

Now to $n^{\prime }s$ law of restitution ${\perp }^{lar}$ to slope applies

Speed after impact

$\to$ combine the two component

$\sqrt{(4{)}^2+(3{)}^2}=\sqrt{16+3}$

$\Rightarrow \sqrt{19}ms$