A 24-ton freight car rolls along a straight track with a speed of 10mph. A 2.0 ton automobile falling vertically lands on the freight car from above.
a) what is the speed of the freight car after the car lands on it?
b) the car then rolls sideways off the freight car. What is the speed of the freight car after the car falls off?
To solve this problem, we can use the law of conservation of momentum. According to this principle, the total momentum of an isolated system remains constant if no external forces act on it.
Let's calculate the total momentum before and after the automobile lands on the freight car.
a) Initially, the momentum of the freight car is given by the formula:
P1 = m1 * v1,
where m1 is the mass of the freight car and v1 is its velocity.
Given:
Mass of the freight car (m1) = 24 tons = 24,000 kg
Velocity of the freight car (v1) = 10 mph
Converting the velocity to meters per second:
1 mph = 0.44704 m/s
So, v1 = 10 mph * 0.44704 m/s = 4.4704 m/s
Plugging the values into the formula:
P1 = 24,000 kg * 4.4704 m/s
P1 = 107,289.6 kg·m/s
The momentum of the automobile before it lands is given by:
P2 = m2 * v2,
where m2 is the mass of the automobile and v2 is its velocity.
Given:
Mass of the automobile (m2) = 2.0 tons = 2,000 kg
Considering it is falling vertically, v2 = 0 m/s (since it was not moving horizontally)
Plugging the values into the formula:
P2 = 2,000 kg * 0 m/s
P2 = 0 kg·m/s
The total momentum before landing is the sum of the momentums of the freight car and the automobile:
P_initial = P1 + P2
P_initial = 107,289.6 kg·m/s + 0 kg·m/s
P_initial = 107,289.6 kg·m/s
After the automobile lands on the freight car, they will move together, so their total momentum will remain the same.
The total momentum after the automobile lands is:
P_final = (m1 + m2) * v_final,
where v_final is the final velocity of both the freight car and the automobile combined.
Plugging in the known values:
P_final = (24,000 kg + 2,000 kg) * v_final
P_final = 26,000 kg * v_final
Since the total momentum remains the same, we have:
P_initial = P_final
107,289.6 kg·m/s = 26,000 kg * v_final
Solving for v_final:
v_final = 107,289.6 kg·m/s / 26,000 kg
v_final ≈ 4.12 m/s
Therefore, the speed of the freight car after the car lands on it is approximately 4.12 m/s.
b) After the car rolls sideways off the freight car, the final momentum of the system is zero. This is because the car has separated from the freight car, and no external forces act on the system.
We can use the same equation as before, where the total momentum after the car falls off is zero:
P_final = (m1 + m2) * v_final
Plugging in the known values:
0 = (24,000 kg + 2,000 kg) * v_final
Solving for v_final:
v_final = 0 / 26,000 kg
v_final = 0 m/s
Therefore, the speed of the freight car after the car falls off is 0 m/s.
To answer these questions, we need to apply the principles of conservation of momentum.
Conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, assuming there are no external forces acting on the system.
Let's tackle each part of the question separately:
a) What is the speed of the freight car after the car lands on it?
First, we need to determine the initial momentum of the freight car and the automobile before the collision.
The momentum of an object is given by the equation:
Momentum = Mass × Velocity
The momentum of the freight car is:
Momentum of freight car = Mass of freight car × Velocity of freight car
Mass of freight car = 24 tons = 24,000 kg (since 1 ton = 1,000 kg)
Velocity of freight car = 10 mph = 4.47 m/s (since 1 mph ≈ 0.447 m/s)
Momentum of freight car = 24,000 kg × 4.47 m/s
Now, let's consider the momentum of the falling automobile. Since it is falling vertically, its initial horizontal velocity does not affect the collision.
Mass of automobile = 2.0 tons = 2,000 kg
Velocity of automobile = 0 m/s (since it is falling vertically)
Momentum of automobile = 2,000 kg × 0 m/s
Now that we have the momenta of both objects before the collision, we can find the total initial momentum:
Total initial momentum = Momentum of freight car + Momentum of automobile
Now let's assume the two objects stick together after the collision and move with a common final velocity (v).
According to the principle of conservation of momentum:
Total initial momentum = Total final momentum
(Momentum of freight car + Momentum of automobile) = (Total mass) × (Final velocity)
Now we can solve for the final velocity of the combined system:
Total final momentum = (Mass of freight car + Mass of automobile) × Final velocity
Substituting the known values:
(24,000 kg × 4.47 m/s + 2,000 kg × 0 m/s) = (24,000 kg + 2,000 kg) × Final velocity
Solving for Final velocity will give us the speed of the freight car after the automobile lands on it.
b) What is the speed of the freight car after the car falls off?
To find the speed of the freight car after the car falls off, we need to consider the conservation of momentum again. This time, only the automobile is moving, and we assume no external forces acting on the system.
The final momentum of the combined system (freight car and automobile) right before the automobile falls off is equal to the momentum of the automobile after it falls off.
The mass of the system (freight car + automobile) remains the same, but the velocity of the automobile changes.
Let's assume the velocity of the automobile after falling off is v2.
Thus, the final momentum of the combined system is:
Final momentum of the combined system = (Mass of freight car + Mass of automobile) × Final velocity
And this final momentum is equal to the momentum of the automobile after it falls off:
Momentum of automobile after falling off = Mass of automobile × Velocity of automobile after falling off
Equating both equations, we can solve for the velocity of the automobile after falling off:
(Mass of freight car + Mass of automobile) × Final velocity = Mass of automobile × Velocity of automobile after falling off
Now we can solve for Velocity of automobile after falling off, which will give us the speed of the freight car after the automobile falls off.
Remember, it is important to consider the correct units and conversions while performing the calculations.