To find the corresponding speed of the raft when the 60 kg swimmer dives horizontally off the 396 kg raft at 7 m/s, we can use the principle of conservation of momentum. The momentum of the system before and after the swimmer dives must remain constant since no external forces are acting on it.
Here's the step-by-step solution:
1. Understand the Initial Conditions:
- The initial velocity of both the swimmer and the raft is zero because both are at rest.
- Therefore, the initial momentum of the system is zero.
2. Calculate the Final Momentum of the Swimmer:
- The swimmer’s mass [tex]\( m_{\text{swimmer}} = 60 \)[/tex] kg.
- The swimmer’s velocity [tex]\( v_{\text{swimmer}} = 7 \)[/tex] m/s.
- Momentum of the swimmer [tex]\( p_{\text{swimmer}} = m_{\text{swimmer}} \times v_{\text{swimmer}} \)[/tex].
- So, [tex]\( p_{\text{swimmer}} = 60 \times 7 = 420 \)[/tex] kg·m/s.
3. Use Conservation of Momentum:
- According to the conservation of momentum, the total momentum of the system before and after the swimmer dives should be equal.
- Initially, the total momentum is 0 (both the swimmer and the raft were at rest).
- Finally, the total momentum is the sum of the swimmer’s momentum and the raft’s momentum.
- Let the raft's speed be [tex]\( v_{\text{raft}} \)[/tex].
- The mass of the raft [tex]\( m_{\text{raft}} = 396 \)[/tex] kg.
4. Set Up the Momentum Equation:
- Since the total initial momentum is zero, the momentum of the swimmer and the raft must balance each other after the swimmer dives:
[tex]\[ 0 = (m_{\text{swimmer}} \times v_{\text{swimmer}}) + (m_{\text{raft}} \times v_{\text{raft}}) \][/tex]
- Rearrange to solve for the raft’s speed:
[tex]\[ m_{\text{swimmer}} \times v_{\text{swimmer}} = - (m_{\text{raft}} \times v_{\text{raft}}) \][/tex]
5. Solve for [tex]\( v_{\text{raft}} \)[/tex]:
[tex]\[ 60 \times 7 = 396 \times v_{\text{raft}} \][/tex]
[tex]\[ 420 = 396 \times v_{\text{raft}} \][/tex]
[tex]\[ v_{\text{raft}} = \frac{420}{396} \][/tex]
[tex]\[ v_{\text{raft}} \approx 1.0606 \text{ m/s} \][/tex]
6. Match the Calculated Speed to the Closest Choice:
- The calculated raft speed [tex]\( \approx 1.0606 \)[/tex] m/s.
- The given choices are: 1.7 m/s, 1.1 m/s, 1.5 m/s, 1.2 m/s.
- The closest choice to [tex]\( 1.0606 \)[/tex] m/s is [tex]\( 1.1 \)[/tex] m/s.
Therefore, the corresponding raft speed is 1.1 m/s.
