To solve this problem, we need to use the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant if no external forces act on it.
Here's the step-by-step solution:
1. Define the variables:
- Mass of the first train ([tex]\( m_1 \)[/tex]) = 5000 kg
- Velocity of the first train ([tex]\( v_1 \)[/tex]) = 100 m/s
- Velocity of the combined trains after the collision ([tex]\( v_f \)[/tex]) = 50 m/s
- We need to find the mass of the second train ([tex]\( m_2 \)[/tex]).
2. Initial momentum:
- Since the second train is initially at rest, its initial velocity ([tex]\( v_2 \)[/tex]) = 0 m/s.
- The initial momentum of the system ([tex]\( P_i \)[/tex]) is the sum of the momenta of the two trains:
[tex]\[
P_i = (m_1 \cdot v_1) + (m_2 \cdot v_2)
\][/tex]
[tex]\[
P_i = (5000 \, \text{kg} \cdot 100 \, \text{m/s}) + (m_2 \cdot 0 \, \text{m/s})
\][/tex]
[tex]\[
P_i = 500,000 \, \text{kg} \cdot \text{m/s}
\][/tex]
3. Final momentum:
- After the collision, the two trains stick together and move as a single entity with a combined mass ([tex]\( m_1 + m_2 \)[/tex]) and a velocity ([tex]\( v_f \)[/tex]).
- The final momentum of the system ([tex]\( P_f \)[/tex]) is:
[tex]\[
P_f = (m_1 + m_2) \cdot v_f
\][/tex]
[tex]\[
P_f = (5000 \, \text{kg} + m_2) \cdot 50 \, \text{m/s}
\][/tex]
4. Conservation of momentum:
- Since the initial and final momenta are equal, we set them equal to each other:
[tex]\[
P_i = P_f
\][/tex]
[tex]\[
500,000 \, \text{kg} \cdot \text{m/s} = (5000 \, \text{kg} + m_2) \cdot 50 \, \text{m/s}
\][/tex]
5. Solve for [tex]\( m_2 \)[/tex]:
[tex]\[
500,000 = (5000 + m_2) \cdot 50
\][/tex]
[tex]\[
500,000 = 250,000 + 50 \cdot m_2
\][/tex]
[tex]\[
250,000 = 50 \cdot m_2
\][/tex]
[tex]\[
m_2 = \frac{250,000}{50}
\][/tex]
[tex]\[
m_2 = 5000 \, \text{kg}
\][/tex]
Therefore, the mass of the second train is [tex]\( 5000 \, \text{kg} \)[/tex].
The correct answer is:
B. [tex]\( 5,000 \, \text{kg} \)[/tex]
