Question

A ball of mass m is dropped on to a floor from certain height, collision is perfectly elastic and the ball rebounds to the same height and again falls. Find the average force exerted by the ball on the floor during a long time interval.

• A. mg
• B. > mg
• C. < mg
• D. none of these

Answer 1

The correct option is A mg

Let the ball is dropped from a point A at height $h$ above the floor and after collision, it reaches to point B which is also at height $h$ above the ground.

Velocity with which the ball reaches the floor $v=\sqrt{2gh}$

Also $h=0+\frac{1}{2}g(t_1{)}^2$ $⟹t_1=\sqrt{\frac{2h}{g}}$

As the collision is perfectly elastic, thus velocity of ball after the collision is also $v$

Similarly $t_2=\sqrt{\frac{2h}{g}}$

Net momentum imparted to the floor due to 1st collision $P_1=mv-(-mv)=2mv=2m\sqrt{2gh}$

Total time taken by ball during the motion from A to B, $T_1=t_1+t_2=2\sqrt{\frac{2h}{g}}$

Thus total momentum imparted to the floor due to n collision $P_T=nP=2mn\sqrt{2gh}$

Total time taken $T=n{T}_1=2n\sqrt{\frac{2h}{g}}$

$\therefore$ total force exerted to the floor $F=\frac{P_T}{T}=\frac{2mn\sqrt{2gh}}{2n\sqrt{\frac{2h}{g}}}$

$⟹$ $F=mg$