Question

A ball of mass $m$ moving with a velocity $v$ hits a massive wall of mass $M(M>>m)$ moving towards the ball with a velocity $u$. An elastic impact lasts for a time $\Delta t$.

• A. The average elastic force acting on the ball is $\frac{m(u+v)}{\Delta t}$
• B. The average elastic force acting on the ball is $\frac{2m(u+v)}{\Delta t}$
• C. The kinetic energy of the ball increases by $2mu(u+v)$
  
• D. The kinetic energy of the ball remans the same after the collision.

Answer 1

Answer: (B), (C)

The correct options are
B The average elastic force acting on the ball is $\frac{2m(u+v)}{\Delta t}$
C The kinetic energy of the ball increases by $2mu(u+v)$



Let $v^{\prime }$ be velocity of the ball after collision, away from the wall.
In an elastic collision
$v_{sep}=v_{app}$
$v^{\prime }-u=v+u$
$v^{\prime }=v+2u$
Change in momentum of ball is $|p_f-p_i$
$=|m(-v^{\prime })-mv|$
$=m(v^{\prime }+v)$
$=2m(u+v)$
average force $=\frac{\Delta p}{\Delta t}=\frac{2m(u+v)}{\Delta t}$
Change in KE = $K_f-K_i=\frac{1}{2}m{v}^{\prime 2}-\frac{1}{2}m{v}^2=\frac{m}{2}\left(\right(v+2u{)}^2-v^2)=\frac{m}{2}\left(\right(2v+2u\left)2u\right)$
The kinetic energy of the ball increases by $2mu(u+v)$