A ball of mass m strikes a rigid walJ with speed u and rebounds with the same speed. The impulse imparted to the ball by the wall is

Options

A. 2mu
B. mu
C. zero
D. `-2"mu"`

Explanation

To solve the problem of finding the impulse imparted to a ball by a rigid wall when it strikes and rebounds, we can follow these steps:

### Step 1: Understand the Initial and Final Momentum
- The ball has a mass \( m \) and strikes the wall with a speed \( u \).
- The initial momentum (\( p_i \)) of the ball before it strikes the wall is given by:
\[ p_i = m \cdot u \]

### Step 2: Determine the Final Momentum
- After striking the wall, the ball rebounds with the same speed \( u \) but in the opposite direction. Therefore, the final momentum (\( p_f \)) is:
\[ p_f = m \cdot (-u) = -m \cdot u \]

### Step 3: Calculate the Change in Momentum
- The impulse (\( J \)) imparted to the ball by the wall is equal to the change in momentum. This can be expressed as:
\[ J = p_f - p_i \]
- Substituting the values we found for initial and final momentum:
\[ J = (-m \cdot u) - (m \cdot u) \]

### Step 4: Simplify the Expression
- Simplifying the expression gives:
\[ J = -m \cdot u - m \cdot u = -2m \cdot u \]

### Step 5: Conclusion
- The impulse imparted to the ball by the wall is:
\[ J = -2m \cdot u \]
- The negative sign indicates that the direction of the impulse is opposite to the direction of the initial momentum.

### Final Answer
The impulse imparted to the ball by the wall is \( -2mu \).

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