100 balls each of mass m moving with speed v simultaneously strike a wall normally and reflected back with same speed, in time t s. The total force exerted by the balls on the wall is

JEE Mains - Force Exerted by Colliding Balls

Question: 100 balls, each of mass \( m \), moving with speed \( v \), simultaneously strike a wall normally and are reflected back with the same speed, all within a time interval of \( t \) seconds. Determine the total force exerted by the balls on the wall.
[31-Jan-2023 Shift 1]

Select the correct answer:

(A) \( \frac{100mv}{t} \)

(B) \( \frac{200mv}{t} \)

(C) \( \frac{400mv}{t} \)

(D) \( \frac{800mv}{t} \)

Solution

Each ball, upon striking the wall, is reflected back with the same speed. Hence, the change in momentum for one ball is: \[ \Delta p = p_{\text{final}} - p_{\text{initial}} = (-mv) - (mv) = -2mv. \]

The magnitude of the change in momentum per ball is \( 2mv \).

For 100 balls, the total change in momentum is: \[ \Delta P = 100 \times 2mv = 200mv. \]

Since this change in momentum occurs in \( t \) seconds, the average force exerted on the wall (by the impulse-momentum theorem) is: \[ F = \frac{\Delta P}{t} = \frac{200mv}{t}. \]

Therefore, the total force exerted by the balls on the wall is: \[ \boxed{\frac{200mv}{t}}. \]