An average force of 900 N is applied to a 400 g steel ball moving at 15 m/s by a collision lasting 30 ms. If the force is in a direction opposite to the initial velocity of the ball then find the magnitude of final momentum of the ball.

Options

A. 6 kg m/s
B. 21 kg m/s
C. 33 kg m/s
D. 27 kg m/s

Answer

To solve the problem, we need to find the final momentum of a steel ball after a force is applied in the opposite direction to its motion. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
- Mass of the steel ball (m) = 400 g = 0.4 kg (convert grams to kilograms by dividing by 1000)
- Initial velocity (u) = 15 m/s
- Average force (F) = 900 N (acting in the opposite direction)
- Time duration of the collision (Δt) = 30 ms = 30 × 10^(-3) s (convert milliseconds to seconds)

### Step 2: Calculate the Initial Momentum
The initial momentum (P_initial) can be calculated using the formula:
\[ P_{\text{initial}} = m \times u \]
Substituting the values:
\[ P_{\text{initial}} = 0.4 \, \text{kg} \times 15 \, \text{m/s} = 6 \, \text{kg m/s} \]

### Step 3: Use Newton's Second Law to Find Change in Momentum
According to Newton's second law, the change in momentum (ΔP) can be expressed as:
\[ F \times \Delta t = P_{\text{final}} - P_{\text{initial}} \]
Rearranging gives us:
\[ P_{\text{final}} = P_{\text{initial}} + F \times \Delta t \]

### Step 4: Calculate the Change in Momentum
Now, we can calculate the change in momentum:
\[ F \times \Delta t = 900 \, \text{N} \times 30 \times 10^{-3} \, \text{s} = 900 \times 0.03 = 27 \, \text{kg m/s} \]

### Step 5: Calculate Final Momentum
Since the force is acting in the opposite direction, we need to subtract the change in momentum from the initial momentum:
\[ P_{\text{final}} = P_{\text{initial}} - (F \times \Delta t) \]
Substituting the values:
\[ P_{\text{final}} = 6 \, \text{kg m/s} - 27 \, \text{kg m/s} = -21 \, \text{kg m/s} \]

### Step 6: Find the Magnitude of Final Momentum
The magnitude of the final momentum is:
\[ |P_{\text{final}}| = 21 \, \text{kg m/s} \]

### Conclusion
The magnitude of the final momentum of the ball is **21 kg m/s**.

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