Question

A ball hits the surface with initial momentum p repeatedly find the momentum transferred to the surface.

Answer 1

The initial momentum of the ball is p:

The momenta of the ball after each successive collision be $p_1 ​, p_2 ​, p_3$ and so on till it comes to rest  

For 1st collision :

$⟹p _ 1 ​ =ep$

Thus net momentum imparted to the floor by 1st collision, 

$P _ 1^ ′ ​ =p _ 1 ​ −(−p)$

$∴ P _ 1^ ′ ​ =ep+p=p(1+e) .............(1)$

For 2nd collision : 

$⟹p_ 2 ​ =ep _ 1 ​ =e ^2 p$

Thus net momentum imparted to the floor by 2nd collision,

$P _ 2 ^′ ​ =p _ 2 ​ −(−p_ 1 ​ )$

$∴ P _ 2 ^′ ​ =e ^2 p+ep=pe(1+e) .............(2)$

Similarly momentum imparted to the floor by each other successive collisions are -

$pe ^ 2 (1+e), pe ^3 (1+e),$ and so on

Total momentum imparted $P=p(1+e)+pe(1+e)+pe ^2 (1+e)+pe ^3 (1+e)+.........∞\ term$

$P=p(1+e)[1+e+e ^2 +e ^3 +............∞]$

$P=p\times\frac{1+e}{1-e}$