Four railroad cars, each of mass 2.50 x 104 kg, are coupled together and coasting along horizontal tracks at some speed vi. Superman, riding on the second car, uncouples the very front car and gives it a big push, increasing its speed in the same direction to 4.00 m/s. The remaining three cars continue moving in the same direction, now at 2.00 m/s.
A railroad car of mass 2.50 x 104 kg is moving with a speed of 4.00 m/s. It collides and
couples with three other coupled railroad cars, each of the same mass as the single car and
moving in the same direction with an initial speed of 2.00 m/s.
(a) What is the speed of the four cars after the collision?
(b) How much mechanical energy is lost in the collision?
To solve this problem, we can use the principle of conservation of momentum. The initial momentum of the system (before Superman pushes the front car) should be equal to the final momentum of the system (after the front car is pushed).
Let's denote the mass of each car as m = 2.50 x 10^4 kg.
1. Before the front car is pushed:
The initial momentum of the system is the sum of the momentum of each car.
Initial momentum = (m × vi) + (m × 0) + (m × 0) + (m × 0) = 4mvi
2. After Superman pushes the front car:
The final momentum of the system is the sum of the momentum of each car after the push.
Final momentum = [(m × (vi - 4m/s))] + [m × (2m/s)] + [m × (2m/s)] + [m × (2m/s)] = 2m(vi - 4m/s) + (3m × 2m/s)
According to the conservation of momentum, the initial momentum should be equal to the final momentum.
4mvi = 2m(vi - 4m/s) + 6m
Simplifying the equation:
4mvi = 2mvi - 8m^2/s + 6m
Moving all terms to one side of the equation:
2mvi = 8m^2/s - 2m
Dividing both sides of the equation by 2m:
vi = 4m/s - 1
Therefore, the initial velocity of the system before the front car is pushed is vi = 4m/s - 1.
To solve this problem, we need to apply the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant if no external forces act on it.
Let's denote the initial velocity of the system (four coupled cars) as "vi." We also need to determine the final velocities of the cars after the separation and the push applied to the first car.
First, let's calculate the total initial momentum of the system. The initial momentum (Pinitial) is the sum of the momentum of each individual car:
Pinitial = m1 * v1 + m2 * v2 + m3 * v3 + m4 * v4
Where:
m1, m2, m3, m4 are the masses of the respective cars, which are all the same (2.50 x 10^4 kg).
v1, v2, v3, v4 are the initial velocities of the respective cars, which are all equal to vi.
So, Pinitial = (2.50 x 10^4 kg)(vi) + (2.50 x 10^4 kg)(vi) + (2.50 x 10^4 kg)(vi) + (2.50 x 10^4 kg)(vi)
Pinitial = 10.0 x 10^4 kg * vi
Next, after Superman uncouples and pushes the first car, its velocity changes to 4.00 m/s. The remaining three cars continue moving with a velocity of 2.00 m/s.
Now, let's calculate the momentum of each car after the separation:
Car 1 (uncoupled): m1 * v1 = (2.50 x 10^4 kg)(4.00 m/s)
Car 2, 3, 4: m2 * v2 + m3 * v3 + m4 * v4 = (2.50 x 10^4 kg)(2.00 m/s) + (2.50 x 10^4 kg)(2.00 m/s) + (2.50 x 10^4 kg)(2.00 m/s)
Finally, the total final momentum (Pfinal) can be calculated as the sum of the momentum of each individual car after the separation:
Pfinal = m1 * v1 + m2 * v2 + m3 * v3 + m4 * v4 = m1 * 4.00 + m2 * 2.00 + m3 * 2.00 + m4 * 2.00
Now, you can substitute the values for mass (2.50 x 10^4 kg) into the equation and solve for Pfinal.