A 55-kg swimmer is standing on a stationary 210 kg boating raft. The swimmer then runs off the raft horizontally with a velocity of +4.6 m/s relative to the shore. Find the recoil velocity that the raft would have if there were no friction and resistance due to the water.

Options

A. `+1.2` m/s
B. `-1.2` m/s
C. `+0.60` m/s
D. `-2.4` m/s

Answer

To solve the problem, we will use the principle of conservation of momentum. Here’s a step-by-step breakdown of the solution:

### Step 1: Understand the system
We have a swimmer with a mass of \( m = 55 \, \text{kg} \) and a raft with a mass of \( M = 210 \, \text{kg} \). Initially, both the swimmer and the raft are at rest.

### Step 2: Write down the initial momentum
Since both the swimmer and the raft are stationary, the initial momentum (\( p_{\text{initial}} \)) of the system is:
\[ p_{\text{initial}} = (m + M) \cdot 0 = 0 \]

### Step 3: Write down the final momentum
When the swimmer runs off the raft with a velocity of \( v_s = +4.6 \, \text{m/s} \) relative to the shore, the raft will have a recoil velocity \( v_r \) in the opposite direction. The final momentum (\( p_{\text{final}} \)) of the system can be expressed as:
\[ p_{\text{final}} = m \cdot v_s + M \cdot v_r \]
Substituting the values, we have:
\[ p_{\text{final}} = 55 \cdot 4.6 + 210 \cdot v_r \]

### Step 4: Apply conservation of momentum
According to the conservation of momentum:
\[ p_{\text{initial}} = p_{\text{final}} \]
Thus, we have:
\[ 0 = 55 \cdot 4.6 + 210 \cdot v_r \]

### Step 5: Solve for the raft's velocity
Now, we can solve for \( v_r \):
\[ 210 \cdot v_r = -55 \cdot 4.6 \]
\[ v_r = \frac{-55 \cdot 4.6}{210} \]
Calculating the right side:
\[ v_r = \frac{-253}{210} \approx -1.2 \, \text{m/s} \]

### Conclusion
The recoil velocity of the raft is approximately \( -1.2 \, \text{m/s} \). The negative sign indicates that the raft moves in the opposite direction to the swimmer.