A railroad diesel engine weighs four times as much has a freight car. if the diesel engine coasts at 5 km/h into a freight car that is initially at rest, show that the speed of the coupled cars is 4 km/h.
4m*5=(4m+m)V
V=5(4/5)=4km/hr
To solve this problem step-by-step, we can use the principle of conservation of momentum.
Step 1: Define the variables
Let's assume:
- m1 = mass of the diesel engine
- m2 = mass of the freight car
- v1 = initial velocity of the diesel engine (5 km/h)
- v2 = initial velocity of the freight car (0 km/h)
- v3 = final velocity of the coupled cars
Step 2: Apply the conservation of momentum
According to the conservation of momentum, the total momentum before the collision is equal to the total momentum after the collision.
Initial momentum = Final momentum
Step 3: Calculate the initial momentum
The initial momentum can be calculated as the product of the mass and velocity of each object:
Initial momentum = (m1 * v1) + (m2 * v2)
= (m1 * 5 km/h) + (m2 * 0 km/h)
= m1 * 5 km/h
Step 4: Calculate the final momentum
The final momentum can be calculated as the product of the mass and velocity of the coupled cars:
Final momentum = (m1 + m2) * v3
Step 5: Apply the conservation of momentum
Based on the conservation of momentum principle (Initial momentum = Final momentum), we have:
m1 * 5 km/h = (m1 + m2) * v3
Step 6: Substitute the given relationship between masses
The problem states that the diesel engine weighs four times as much as the freight car:
m1 = 4 * m2
Substituting this relationship into the equation:
4 * m2 * 5 km/h = (4 * m2 + m2) * v3
20 * m2 km/h = 5 * m2 * 4 km/h + m2 * v3
20 * m2 km/h = 20 * m2 km/h + m2 * v3
20 * m2 km/h - 20 * m2 km/h = m2 * v3
0 = m2 * v3
Step 7: Determine the final velocity
Since we have 0 = m2 * v3, we can conclude that the final velocity v3 must be zero.
Step 8: Convert the final velocity to km/h
The final velocity of 0 is equivalent to 0 km/h.
Therefore, the speed of the coupled cars is 0 km/h, indicating that they come to a stop after the collision.
To solve this problem, we can apply the principle of conservation of momentum. The momentum before the collision is equal to the momentum after the collision.
The momentum of an object is given by the product of its mass and velocity. Let's denote the mass of the diesel engine as M and the mass of the freight car as m. Given that the diesel engine weighs four times as much as the freight car, we have M = 4m.
Now, let's consider the initial conditions. The diesel engine has a mass M and it is coasting at 5 km/h, while the freight car has a mass m and is initially at rest (0 km/h).
The initial momentum of the diesel engine is calculated as P_initial_engine = M * V_initial_engine, where V_initial_engine is the initial velocity of the diesel engine.
In this case, P_initial_engine = M * 5 km/h.
The initial momentum of the freight car is calculated as P_initial_freight_car = m * V_initial_freight_car, where V_initial_freight_car is the initial velocity of the freight car.
In this case, P_initial_freight_car = m * 0 km/h, as the freight car is at rest initially.
According to the principle of conservation of momentum, the total initial momentum (P_initial_total) is equal to the final momentum (P_final_total):
P_initial_total = P_final_total
P_initial_engine + P_initial_freight_car = P_final_engine + P_final_freight_car
M * V_initial_engine + m * V_initial_freight_car = (M + m) * V_final_coupled_cars
Substituting the given values and simplifying the equation:
4m * 5 km/h + m * 0 km/h = (4m + m) * 4 km/h
20m = 5m * 4
20 = 5 * 4
20 = 20
This equation shows that the initial momentum of the diesel engine is equal to the final momentum of the coupled cars. Therefore, the speed of the coupled cars (V_final_coupled_cars) is 4 km/h.