91Ó°ÊÓ

When dealing with elastic collisions, understanding momentum change is crucial. Momentum is defined as the product of an object's mass and its velocity, expressed as \( p = mv \). In an elastic collision, an object like a ball will bounce off a surface with the same speed it had before hitting it. The direction, however, changes—which is key to finding the momentum change.

Consider a ball moving towards a surface at velocity \( u \). After striking the surface elastically, it will rebound with the same velocity, but in the opposite direction. Therefore, its velocity changes from \( u \) to \( -u \). The change in momentum for the ball is thus given by:

\[ \Delta p = m(-u) - mu = -2mu \]

This equation shows that the total change in momentum comprises both the initial and final momentum. As it illustrates, the factor of \(-2\) signifies the complete inversion of its velocity upon collision, characteristic of elastic collisions.

The Impulse-Momentum Principle connects the impulse experienced by an object to its change in momentum. Impulse itself is defined as the force applied multiplied by the time duration over which it is applied, typically written as:

\[ I = F \times \Delta t \]

In the context of our problem, the impulse is associated with the total change in momentum experienced by the surface as the balls strike it. For each ball, this impulse results from its momentum change \(-2mu\). When \( n \) balls hit the surface every second, the cumulative impulse becomes:

\[ I_{total} = n(-2mu) = -2mnu \]

Since we're dealing with a time interval of 1 second (each second these collisions happen), the impulse simplifies to this total change in momentum over this duration. Therefore, using the principle, the average force exerted by the surface can be calculated.

Calculating the average force on a surface due to repetitive events like collisions involves both understanding of momentum change and impulse. Here, the average force \( F \) can be derived using the impulse-momentum principle. Given that the impulse \( I \) over time \( \Delta t = 1 \) second is the change in momentum:

\[ F = \frac{I_{total}}{\Delta t} = \frac{-2mnu}{1} = -2mnu \]

Since the force direction is considered towards where the balls come from, we usually take the magnitude for practical purposes. Thus, the average force experienced by the surface is:

\[ F = 2mnu \]

This highlights how momentum change and the inherent rebound characteristics of elastic collisions translate to a measurable average force. Upon examining the choices in the problem, option \( b \) accurately reflects this force magnitude.