91Ó°ÊÓ

Momentum is a measure of the amount of motion an object has and is defined as the product of an object's mass and its velocity. Mathematically, it is given by the formula:
\( p = mv \)
When a ball hits the door and bounces back, this changes its momentum because the direction of the velocity reverses.
Let's consider a ball with mass \( m \) traveling at speed \( v \) before hitting the door. The initial momentum of the ball is
\[ p_{\text{initial}} = m \times v = mv \]
After the ball hits the door and bounces back, its speed remains \( v \), but the direction reverses, making its momentum: \[ p_{\text{final}} = m \times (-v) = -mv \]
The change in momentum is calculated by subtracting the initial momentum from the final momentum:
\[ \text{Change in momentum} = p_{\text{final}} - p_{\text{initial}} = -mv - mv = -2mv \]
Thus, the overall change in momentum for the ball is \( -2mv \). This negative sign indicates the direction of the change opposite to the initial motion.

Force is related to the change in momentum by Newton's second law of motion. The average force exerted on the ball by the door can be calculated from the change in momentum over the time interval \( \text{ \Delta \ t } \):
According to Newton's second law:
\[ F_{\text{avg}} = \frac{\text{Change in momentum}}{\text{ \Delta \ t}} \]
Using the previously calculated change in momentum \( -2mv \), we get: \[ F_{\text{avg}} = \frac{-2mv}{\text{ \Delta \ t}} \]
The average force is therefore:

• Proportional to the mass of the ball ( \( m \) ).
• Proportional to the speed of the ball ( \( v \) ).
• Inversely proportional to the contact time ( \( \text{ \Delta \ t } \) ).

This means for a given mass and speed, a shorter contact time results in a larger average force, and a longer contact time results in a smaller average force.

Newton's second law of motion states that the force exerted on an object is equal to the rate of change of its momentum. This law is expressed as:
\[ F = \frac{dp}{dt} \]
Where

• \( F \) is the force.
• \( dp \) is the change in momentum.
• \( dt \) is the change in time.

In our problem, we see a practical application of this law when the student throws the superball to shut the door. The collision causes a rapid change in the ball's momentum and therefore a significant force.
If we compare it with a clay ball, which sticks to the door, the resulting momentum change is only \( -mv \). This means:
\[ \text{Change in momentum for clay ball} = \text{initial} - \text{final} = mv - 0 = -mv \]
This results in a smaller average force: \[ F_{\text{avg\text{ for clay}} = \frac{-mv}{\text{ \Delta \ t}} \]
As a result, the superball creates a larger force impact, shutting the door more effectively. Therefore, choosing a superball over clay proves more efficient in exerting a larger average force for a swift door closure.