Question

A ball is projected from point O on the ground. It hits a smooth vertical wall AB at a height h and rebounds elastically. The ball finally lands at a point C on the ground. During the course of motion, the maximum height attained by the ball is H. The ratio $\frac{H}{h}$ if $\frac{OA}{OC}=\frac{1}{3}$ is (Answer upto two digits after the decimal point)

Answer 1

In elastic collision, the horizontal component of velocity will get inverted. There is no change in vertical motion of the projectile [as wall is smooth]
In the absence of wall, the path of projectile have been $DEF$. After collision , the path $DGC$ (mirror image of $DEF$)
Time of flight, $T=\frac{2u_y}{g}$
Max height attained during the flight, $H=\frac{u_y^2}{2g}$
Time of flight from O to D is $t=\frac{T}{5}=\frac{2u_y}{5g}$
[$\because$ Horizontal dispalcement is $\frac{1}{5}$ of the range of a usual projectile as can be understood from the diagram]
$\therefore h=u_{y}t-\frac{1}{2}g{t}^2=u_y\frac{2u_y}{5g}-\frac{1}{2}g{\left[\frac{2u_y}{5g}\right]}^2$
$\Rightarrow h=\frac{u_y^2}{2g}[\frac{4}{5}-\frac{4}{25}]=H\times \frac{16}{25}$
$\Rightarrow \frac{H}{h}=\frac{25}{16}=1.5625$