A railroad car with a mass of 1.99 104 kg moving at 2.80 m/s joins with two railroad cars already joined together, each with the same mass as the single car and initially moving in the same direction at 1.52 m/s.
(a) What is the speed of the three joined cars after the collision
(b) What is the decrease in kinetic energy during the collision?
To solve this problem, we can use the principle of conservation of momentum and the principle of conservation of kinetic energy.
(a) The principle of conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision. Momentum is given by the equation:
momentum = mass * velocity
Before the collision:
The single car has a mass of 1.99 × 10^4 kg and a velocity of 2.80 m/s.
The two other cars have the same mass (1.99 × 10^4 kg) each and a velocity of 1.52 m/s.
Total momentum before the collision:
= (mass of single car * velocity of single car) + (mass of first joined car * velocity of first joined car) + (mass of second joined car * velocity of second joined car)
= (1.99 × 10^4 kg * 2.80 m/s) + (1.99 × 10^4 kg * 1.52 m/s) + (1.99 × 10^4 kg * 1.52 m/s)
After the collision:
The three cars will move together as a single unit with a common velocity (let's call it V).
Total momentum after the collision:
= total mass * velocity of three joined cars
= (3 * mass of single car) * V
= (3 * 1.99 × 10^4 kg) * V
Using the principle of conservation of momentum, we can set the total momentum before the collision equal to the total momentum after the collision:
(1.99 × 10^4 kg * 2.80 m/s) + (1.99 × 10^4 kg * 1.52 m/s) + (1.99 × 10^4 kg * 1.52 m/s) = (3 * 1.99 × 10^4 kg) * V
Now we can solve the equation for V to find the velocity of the three joined cars after the collision.
(b) The decrease in kinetic energy during the collision can be calculated by taking the difference between the initial kinetic energy and the final kinetic energy.
Initial kinetic energy:
= 1/2 * mass of single car * (velocity of single car)^2
= 1/2 * (1.99 × 10^4 kg) * (2.80 m/s)^2
Final kinetic energy:
= 1/2 * total mass * (velocity of three joined cars)^2
= 1/2 * (3 * mass of single car) * V^2
The decrease in kinetic energy during the collision is the difference between the initial and final kinetic energy:
Decrease in kinetic energy = Initial kinetic energy - Final kinetic energy
Now that we have set up the equations, we can plug in the values and solve for the desired quantities.