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A [tex]$5,000 \, \text{kg}$[/tex] train is traveling at a velocity of [tex]$100 \, \text{m/s}$[/tex] and hits another train. The two trains stick together, and the new velocity is [tex]$50 \, \text{m/s}$[/tex]. What is the mass of the second train?

A. [tex]$8,000 \, \text{kg}$[/tex]
B. [tex]$10,000 \, \text{kg}$[/tex]
C. [tex]$15,000 \, \text{kg}$[/tex]
D. [tex]$5,000 \, \text{kg}$[/tex]

Sagot :

GinnyAnswer
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]
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