A railroad diesel engine weighs 3.9 times as much as a freight car. The diesel engine coasts at 5.3 km/h into the freight car that is initially at rest. After they couple together, what is their speed?
To find the speed of the coupled diesel engine and freight car, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
Let's denote the initial masses of the diesel engine and the freight car as M1 and M2, and their initial velocities as V1 and V2, respectively. Since the freight car is initially at rest, V2 is 0 km/h.
Now, let's solve the problem step by step:
Step 1: Set up the equation for conservation of momentum before the collision:
M1 * V1 = (M1 + M2) * Vf
where Vf is the final velocity of the coupled system.
Step 2: Use the given information to determine the relationship between the masses of the diesel engine and the freight car:
M1 = 3.9 * M2
Step 3: Substitute the relationship from step 2 into the conservation of momentum equation from step 1:
(3.9 * M2) * V1 = (3.9 * M2 + M2) * Vf
Step 4: Simplify the equation:
(3.9 * V1) * M2 = 4.9 * M2 * Vf
Step 5: Cancel out M2 on both sides of the equation:
3.9 * V1 = 4.9 * Vf
Step 6: Solve for Vf:
Vf = (3.9 * V1) / 4.9
Step 7: Substitute the given value of V1 (5.3 km/h) into the equation:
Vf = (3.9 * 5.3) / 4.9
Step 8: Calculate the value of Vf:
Vf = 4.207 km/h
Thus, the speed of the coupled diesel engine and freight car is approximately 4.207 km/h.