To determine the total momentum of the system before and after the collision of two train cars, we need to follow these steps:
### Step 1: Define the given variables
- Mass of train car 1: [tex]\( m_1 = 5000 \)[/tex] kg
- Velocity of train car 1: [tex]\( v_1 = 10 \)[/tex] m/s
- Mass of train car 2: [tex]\( m_2 = 7000 \)[/tex] kg
- Velocity of train car 2: [tex]\( v_2 = -5 \)[/tex] m/s (negative because it is moving toward car 1)
### Step 2: Calculate the momentum of each train car before the collision
The momentum [tex]\( p \)[/tex] of an object is calculated using the formula:
[tex]\[ p = m \times v \][/tex]
For train car 1:
[tex]\[ p_1 = m_1 \times v_1 = 5000 \, \text{kg} \times 10 \, \text{m/s} = 50000 \, \text{kg} \cdot \text{m/s} \][/tex]
For train car 2:
[tex]\[ p_2 = m_2 \times v_2 = 7000 \, \text{kg} \times (-5) \, \text{m/s} = -35000 \, \text{kg} \cdot \text{m/s} \][/tex]
### Step 3: Calculate the total momentum before the collision
The total momentum of the system before the collision is the sum of the momenta of both train cars:
[tex]\[ \text{Total momentum before} = p_1 + p_2 = 50000 \, \text{kg} \cdot \text{m/s} + (-35000) \, \text{kg} \cdot \text{m/s} = 15000 \, \text{kg} \cdot \text{m/s} \][/tex]
### Step 4: Apply the law of conservation of momentum
According to the law of conservation of momentum, the total momentum of a closed system remains constant if no external forces act on it. This means that the total momentum before the collision must equal the total momentum after the collision.
Therefore,
[tex]\[ \text{Total momentum after} = \text{Total momentum before} = 15000 \, \text{kg} \cdot \text{m/s} \][/tex]
### Summary of Results
- The total momentum of the system before the train cars collide is [tex]\( 15000 \, \text{kg} \cdot \text{m/s} \)[/tex]
- The total momentum of the system must be [tex]\( 15000 \, \text{kg} \cdot \text{m/s} \)[/tex] after the train cars collide.
So, the answers to the questions are:
1. The total momentum of the system before the train cars collide is [tex]\( 15000 \, \text{kg} \cdot \text{m/s} \)[/tex].
2. The total momentum of the system after the train cars collide must be [tex]\( 15000 \, \text{kg} \cdot \text{m/s} \)[/tex].
