A railroad car of mass 2.3*10^4 kg moving at 4.00 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?
(b) How much kinetic energy is lost in the collision?
i want answer please
To solve this problem, we can apply the law of conservation of momentum, which states that the total momentum before a collision is equal to the total momentum after the collision. To find the speed of the three coupled cars after the collision, we need to calculate the initial and final momentum of the system.
Let's break down the problem step by step:
Step 1: Calculate the initial momentum.
The initial momentum is the sum of the individual momenta of each car before the collision. We can calculate it using the equation:
Initial momentum = (mass of car 1) * (velocity of car 1) + (mass of car 2) * (velocity of car 2) + (mass of car 3) * (velocity of car 3)
Given:
Mass of each car (m) = 2.3 * 10^4 kg
Velocity of car 1 (v1) = 4.00 m/s
Velocity of car 2 (v2) = 1.20 m/s
Velocity of car 3 (v3) = 1.20 m/s (since it is moving in the same direction as car 2)
Substituting the values into the equation:
Initial momentum = (2.3 * 10^4 kg * 4.00 m/s) + (2.3 * 10^4 kg * 1.20 m/s) + (2.3 * 10^4 kg * 1.20 m/s)
Initial momentum = 2.3 * 10^4 kg * (4.00 m/s + 1.20 m/s + 1.20 m/s)
Step 2: Calculate the final momentum.
The final momentum is the sum of the individual momenta of the three coupled cars after the collision. Since the cars couple, they move together with the same velocity.
Given:
Final velocity (vf) = ?
Total mass (m1 + m2 + m3) = (m + m + m) = 3m
Using the law of conservation of momentum, we can write:
Initial momentum = Final momentum
(2.3 * 10^4 kg * (4.00 m/s + 1.20 m/s + 1.20 m/s)) = (3m) * vf
Now we can solve for vf:
vf = (2.3 * 10^4 kg * (4.00 m/s + 1.20 m/s + 1.20 m/s)) / (3 * (2.3 * 10^4 kg))
Step 3: Calculate the kinetic energy lost in the collision.
To determine the kinetic energy lost in the collision, we need to find the difference in kinetic energy between the initial and final states. The formula for kinetic energy is given by:
Kinetic energy = (1/2) * mass * velocity^2
First, we calculate the initial kinetic energy:
Initial kinetic energy = (1/2) * (2.3 * 10^4 kg) * (4.00 m/s)^2 + (1/2) * (2.3 * 10^4 kg) * (1.20 m/s)^2 + (1/2) * (2.3 * 10^4 kg) * (1.20 m/s)^2
Next, we calculate the final kinetic energy by substituting the final velocity (vf) into the kinetic energy equation:
Final kinetic energy = (1/2) * (3m) * vf^2
Finally, we can calculate the kinetic energy lost in the collision by taking the difference between the initial and final kinetic energies:
Kinetic energy lost = Initial kinetic energy - Final kinetic energy
By following these steps, you can find the answers to both parts (a) and (b) of the problem.