To solve the problem, we will use the principle of conservation of momentum. Let's break it down step by step.
### Step-by-Step Solution:
1. **Identify the Initial Conditions:**
- The cart has a mass of \( m_c = 100 \, \text{kg} \).
- Each boy has a mass of \( m_b = 20 \, \text{kg} \).
- There are 5 boys, so the total mass of the boys is \( 5 \times 20 = 100 \, \text{kg} \).
- The total mass of the system (cart + boys) is \( m_{\text{total}} = m_c + 100 \, \text{kg} = 200 \, \text{kg} \).
- The initial speed of the cart is \( v_0 = 10 \, \text{m/s} \).
2. **Calculate the Initial Momentum:**
- The initial momentum \( p_{\text{initial}} \) of the system is given by:
\[
p_{\text{initial}} = m_{\text{total}} \times v_0 = 200 \, \text{kg} \times 10 \, \text{m/s} = 2000 \, \text{kg m/s}
\]
3. **Consider the First Boy Jumping Off:**
- When the first boy jumps off with a speed of \( 2 \, \text{m/s} \) in the opposite direction to the cart's motion, his velocity with respect to the ground becomes:
\[
v_{\text{boy}} = v_0 - 2 = 10 \, \text{m/s} - 2 \, \text{m/s} = 8 \, \text{m/s} \, \text{(in the negative direction)}
\]
4. **Calculate the Final Momentum After the Boy Jumps:**
- After the boy jumps, the mass of the cart becomes \( 100 \, \text{kg} + 80 \, \text{kg} = 180 \, \text{kg} \) (since there are now 4 boys left).
- Let \( v_f \) be the final speed of the cart after the boy jumps off. The final momentum \( p_{\text{final}} \) is given by:
\[
p_{\text{final}} = (m_c + 80) v_f + m_b v_{\text{boy}} = 180 v_f + 20(-8)
\]
- This simplifies to:
\[
p_{\text{final}} = 180 v_f - 160
\]
5. **Set Initial Momentum Equal to Final Momentum:**
- By the conservation of momentum:
\[
p_{\text{initial}} = p_{\text{final}}
\]
\[
2000 = 180 v_f - 160
\]
6. **Solve for \( v_f \):**
- Rearranging the equation gives:
\[
180 v_f = 2000 + 160
\]
\[
180 v_f = 2160
\]
\[
v_f = \frac{2160}{180} = 12 \, \text{m/s}
\]
### Final Answer:
The speed of the cart after the first boy has jumped out is \( 12 \, \text{m/s} \).
