A railroad car of mass 2.15 ✕ 104 kg moving at 3.15 m/s collides and couples with two coupled railroad cars, each of the same mass as the single car and moving in the same direction at 1.20 m/s.
(a) What is the speed of the three coupled cars after the collision?
____________ m/s
(b) How much kinetic energy is lost in the collision?
____________ J
a. M1*V1 + M2*V2 + M3*V3=M1V+M2V+M3V
21500*3.15+21500*1.2+21500*1.2 = 64,500V.
116,100 = 64,500V, V = 1.6 m/s.
b. KE1 = 0.5M*V1^2+0.5M*V2^2+0.5M*V3^2.
KE1 = M(0.5V1^2+0.5V2^2+0.5V3^2).
KE1 = 21,500(4.96 + 0.72 + 0.72) = 137,600 J.
KE2 = 0.5M*V^2 = 32,250*1.8^2 = 105,300 J.
KE!-KE2 = 137,600 - 105,300 = 32,300 J. lost.
A: Well, it seems like these railroad cars had a little crash party. After the collision, when they all decide to couple up, their speed becomes a team effort. To calculate the speed, we have to take into account the momentum of each car. So, let's get into locomotion mode and calculate!
The initial momentum of the single car is given by:
P_initial = mass_single * velocity_single
P_initial = (2.15 ✕ 10^4 kg) * (3.15 m/s)
Since the other two cars are of the same mass and moving at the same velocity, their individual momentums are:
P_initial_cars = mass_cars * velocity_each_car
P_initial_cars = (2.15 ✕ 10^4 kg) * (1.20 m/s)
Now, in the end, when they all couple up, their momentum becomes:
P_final = P_initial + P_initial_cars + P_initial_cars
P_final = (2.15 ✕ 10^4 kg) * (3.15 m/s) + (2.15 ✕ 10^4 kg) * (1.20 m/s) + (2.15 ✕ 10^4 kg) * (1.20 m/s)
Divide the final momentum by the total mass of the three cars to find the speed:
Final velocity = P_final / (3 * mass_single)
Final velocity = [(2.15 ✕ 10^4 kg) * (3.15 m/s) + (2.15 ✕ 10^4 kg) * (1.20 m/s) + (2.15 ✕ 10^4 kg) * (1.20 m/s)] / (3 * (2.15 ✕ 10^4 kg))
Final velocity = ________ m/s (You complete the calculation)
B: Now, let's put on our detective hats and investigate how much kinetic energy went missing during this railroad collision. We'll need the initial kinetic energy and the final kinetic energy to figure out the difference.
The initial kinetic energy can be found using the formula:
KE_initial = (1/2) * mass_single * velocity_single^2
KE_initial = (1/2) * (2.15 ✕ 10^4 kg) * (3.15 m/s)^2
The final kinetic energy, after the collision, can be calculated using:
KE_final = (1/2) * (3 * mass_single) * final_velocity^2
KE_final = (1/2) * (3 * 2.15 ✕ 10^4 kg) * (final_velocity)^2
The lost kinetic energy can then be determined as:
Lost KE = KE_initial - KE_final
Lost KE = [(1/2) * (2.15 ✕ 10^4 kg) * (3.15 m/s)^2] - [(1/2) * (3 * 2.15 ✕ 10^4 kg) * (final_velocity)^2]
Now it's time to crack this case and calculate the lost kinetic energy:
Lost KE = ________ J (You complete the calculation)
Remember, every train wreck has some losses. Stay safe and mind the tracks!
To solve this problem, we can use the law of conservation of momentum and the law of conservation of kinetic energy.
(a) First, let's calculate the momentum of each car before the collision and after the collision.
The momentum of an object is given by the formula: momentum = mass * velocity.
For the single car before the collision:
Momentum1 = mass1 * velocity1 = (2.15 ✕ 10^4 kg) * (3.15 m/s)
For the two coupled cars before the collision:
Momentum2 = mass2 * velocity2 = (2 * 2.15 ✕ 10^4 kg) * (1.20 m/s)
Now, let's calculate the total momentum before the collision:
Total momentum before collision = Momentum1 + Momentum2
After the collision, the three coupled cars will move together as one unit, so their total mass will be the sum of the masses of each individual car.
Let's denote the mass of each car as "m":
Total mass after collision = m + m + m = 3m
Let's denote the speed of the coupled cars after the collision as "v":
Total momentum after collision = total mass after collision * speed after collision
= (3m) * v
According to the law of conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision:
Momentum1 + Momentum2 = (3m) * v
Now, we can solve for "v" by substituting the given values:
(2.15 ✕ 10^4 kg) * (3.15 m/s) + (2 * 2.15 ✕ 10^4 kg) * (1.20 m/s) = (3m) * v
Solve this equation to find the value of "v", which will be the speed of the three coupled cars after the collision.
(b) To calculate the kinetic energy lost in the collision, we need to find the initial kinetic energy before the collision and the final kinetic energy after the collision.
The kinetic energy is given by the formula: kinetic energy = 0.5 * mass * velocity^2
For the single car before the collision:
Initial kinetic energy1 = 0.5 * mass1 * velocity1^2
For the two coupled cars before the collision:
Initial kinetic energy2 = 0.5 * mass2 * velocity2^2
Total initial kinetic energy = Initial kinetic energy1 + Initial kinetic energy2
For the three coupled cars after the collision:
Final kinetic energy = 0.5 * total mass after collision * speed after collision^2
The kinetic energy lost in the collision is the difference between the total initial kinetic energy and the final kinetic energy:
Kinetic energy lost = Total initial kinetic energy - Final kinetic energy
Now, substitute the given values into the equations and calculate the kinetic energy lost in joules.
Note: Make sure to convert the masses, velocities, and speeds to standard SI units before performing the calculations.