To determine the mass of the second train, we can use the principle of conservation of momentum. The principle states that the total momentum before a collision is equal to the total momentum after the collision, provided no external forces act on the system.
Let's break this down step by step:
1. Identify the known quantities:
- Mass of the first train ([tex]\( m_1 \)[/tex]): [tex]\( 5,000 \)[/tex] kg
- Initial velocity of the first train ([tex]\( v_1 \)[/tex]): [tex]\( 100 \)[/tex] m/s
- Combined velocity after collision ([tex]\( v_{\text{final}} \)[/tex]): [tex]\( 50 \)[/tex] m/s
2. Define the unknown quantity:
- Mass of the second train ([tex]\( m_2 \)[/tex])
3. Write the conservation of momentum equation:
The equation for the conservation of momentum before and after the collision is:
[tex]\[
m_1 \cdot v_1 = (m_1 + m_2) \cdot v_{\text{final}}
\][/tex]
4. Substitute the known values into the equation:
[tex]\[
5000 \ \text{kg} \cdot 100 \ \text{m/s} = (5000 \ \text{kg} + m_2) \cdot 50 \ \text{m/s}
\][/tex]
5. Solve for the mass of the second train ([tex]\( m_2 \)[/tex]):
[tex]\[
500,000 \ \text{kg} \cdot \text{m/s} = (5000 \ \text{kg} + m_2) \cdot 50 \ \text{m/s}
\][/tex]
6. Divide both sides by [tex]\( 50 \ \text{m/s} \)[/tex] to isolate [tex]\( m_2 \)[/tex]:
[tex]\[
10,000 \ \text{kg} = 5000 \ \text{kg} + m_2
\][/tex]
7. Subtract the mass of the first train from both sides to solve for [tex]\( m_2 \)[/tex]:
[tex]\[
10,000 \ \text{kg} - 5000 \ \text{kg} = m_2
\][/tex]
[tex]\[
m_2 = 5000 \ \text{kg}
\][/tex]
Therefore, the mass of the second train is [tex]\( 5,000 \)[/tex] kg.
The correct answer is:
D. [tex]$5,000 kg$[/tex]
