To determine the combined velocity of the railroad cars immediately after the collision, we can use the principle of conservation of momentum. Before the collision, only the first car is moving, while the second car is stationary.
The formula for momentum is given by:
\[ \text{Momentum} = \text{mass} \times \text{velocity} \]
Step 1: Calculate the momentum before the collision
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For the first car (mass \( m_1 = 6000 \) kg, velocity \( v_1 = 5 \) m/s): \[ p_1 = m_1 \times v_1 = 6000 , \text{kg} \times 5 , \text{m/s} = 30000 , \text{kg m/s} \]
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For the second car (mass \( m_2 = 4000 \) kg, velocity \( v_2 = 0 \) m/s): \[ p_2 = m_2 \times v_2 = 4000 , \text{kg} \times 0 , \text{m/s} = 0 , \text{kg m/s} \]
Step 2: Total momentum before the collision
\[ \text{Total momentum before} = p_1 + p_2 = 30000 , \text{kg m/s} + 0 , \text{kg m/s} = 30000 , \text{kg m/s} \]
Step 3: Calculate the combined mass after the collision
After the collision, the two cars couple together, so the total mass \( m_f \) is:
\[ m_f = m_1 + m_2 = 6000 , \text{kg} + 4000 , \text{kg} = 10000 , \text{kg} \]
Step 4: Set the total momentum after the collision equal to the total momentum before the collision
Let \( v_f \) be the combined velocity immediately after the collision. According to the conservation of momentum:
\[ \text{Total momentum after} = m_f \times v_f \]
Setting this equal to the total momentum before the collision:
\[ 30000 , \text{kg m/s} = 10000 , \text{kg} \times v_f \]
Step 5: Solve for \( v_f \)
\[ v_f = \frac{30000 , \text{kg m/s}}{10000 , \text{kg}} = 3 , \text{m/s} \]
Thus, the combined velocity of the railroad cars immediately after the impact is:
\[ \boxed{3 , \text{m/s}} \]