A freight car with a mass of 29,000 kg rolls down an inclined track through a vertical distance of 2.8 m. At the bottom of the incline, on a level track, the car collides and couples with an identical freight car that was at rest. What percentage of the initial kinetic energy is lost in the collision?
To determine the percentage of the initial kinetic energy lost in the collision, we need to calculate the initial kinetic energy and final kinetic energy of the system.
First, let's find the initial kinetic energy of the system when the freight car travels down the inclined track. The formula for kinetic energy is:
Kinetic energy (KE) = 0.5 * mass * velocity^2
Given:
Mass of one freight car (m) = 29,000 kg
Vertical distance traveled (h) = 2.8 m
We need to find the velocity (v) of the freight car at the bottom of the inclined track. We can use the principle of conservation of energy. The potential energy the car possesses at the top of the incline will convert to kinetic energy at the bottom, neglecting any losses due to friction or other external forces.
Potential energy (PE) = m * g * h
where g is the acceleration due to gravity (9.8 m/s^2)
Let's calculate the potential energy (PE):
PE = 29,000 kg * 9.8 m/s^2 * 2.8 m = 747,440 J
Since PE is converted into KE at the bottom of the incline, the initial kinetic energy (KE_initial) is equal to the potential energy:
KE_initial = PE = 747,440 J
Next, after the collision, the two freight cars couple together and move as one unit with a final velocity (V_final).
We can calculate the velocity of the two-coupled freight cars using the principle of conservation of momentum.
Since the cars have the same mass and one was initially at rest, the total momentum before the collision is zero. Hence, the total momentum after the collision will also be zero.
momentum_before = momentum_after
(mass * velocity_initial) + (mass * 0) = (2 * mass * V_final)
Substituting the values, we get:
29,000 kg * velocity_initial = 2 * 29,000 kg * V_final
velocity_initial = 2 * V_final
Now, let's denote the final velocity of the coupled freight cars as V_final. Combining the above equation with the equation for kinetic energy:
KE_final = (2 * mass * V_final^2)/2
Simplifying, we get:
KE_final = mass * V_final^2
Finally, we can find the percentage of initial kinetic energy lost in the collision by calculating:
Percentage of energy lost = ((KE_initial - KE_final) / KE_initial) * 100
Substituting the known values:
Percentage of energy lost = ((747,440 J - mass * V_final^2) / (747,440 J)) * 100
This equation gives us the percentage of initial kinetic energy lost in the collision.
Please note that we need the final velocity (V_final) to obtain the exact value, and for that, we need additional information or assumptions about the collision, such as whether it is elastic or inelastic.