Question

A ball released from a height $h_0$ above a horizontal surface rebounds to a height $h_1$ after one bounce. The graph that relates to $h_0$ to $h_1$ is shown in Fig. If the ball (of the mass $m$) was dropped from an initial height $h$ and made three bounces, the kinetic energy of the ball immediately after the third impact with the surface was

• A. $(0.8{)}^{3}mgh$
• B. $(0.8{)}^{2}mgh$
• C. ${0.8}^{3}mg\left(h/3\right)$
• D. $[1-(0.8{)}^3]mgh$

Answer 1

The correct option is A $(0.8{)}^{3}mgh$

We know that

$h_n=e^{2n}{h}_0$ (where $e$ is the coff of restitution)for $n=1$

$n_1=e^2{h}_0$

$e^2=\frac{h_1}{h_0}=\frac{80}{100}=0.8.....\left(2\right)$

heigh gain ball after third impact

$h_3=e^6{h}_0$

$=(e^2{)}^3{h}_0$

$h_3=(0.8{)}^3{h}_0$

kinectic energy after this impact = potential energy at $h_3$

$K.E=mg{h}_3$

$K.E=(0.8{)}^{3}mg{h}_0$