A mine car (mass = 452 kg) rolls at a speed of 0.356 m/s on a horizontal track, as the drawing shows. A 179-kg chunk of coal has a speed of 0.863 m/s when it leaves the chute. Determine the velocity of the car/coal system after the coal has come to rest in the car.
To find the velocity of the car/coal system after the coal has come to rest in the car, we can apply the law of conservation of momentum. According to this principle, the total momentum before an event is equal to the total momentum after the event, assuming no external forces are acting on the system.
The momentum of an object is given by the product of its mass and velocity. Mathematically, it can be represented as:
Momentum = Mass × Velocity
Using this equation, we can determine the initial momentum of the car and coal separately and then find the final velocity of the combined system.
Given:
Mass of the car (m1) = 452 kg
Velocity of the car (v1) = 0.356 m/s
Mass of the coal (m2) = 179 kg
Velocity of the coal (v2) = 0.863 m/s
Step 1: Calculate the initial momentum of the car (p1 = m1 * v1):
p1 = 452 kg * 0.356 m/s = 160.912 kg·m/s
Step 2: Calculate the initial momentum of the coal (p2 = m2 * v2):
p2 = 179 kg * 0.863 m/s = 154.677 kg·m/s
Step 3: Calculate the total initial momentum of the system (p_initial = p1 + p2):
p_initial = p1 + p2 = 160.912 kg·m/s + 154.677 kg·m/s = 315.589 kg·m/s
Step 4: Since the coal comes to rest in the car, the final velocity of the coal is 0 m/s. Therefore, the final momentum of the coal (p2_final) is 0 kg·m/s.
Step 5: Apply the principle of conservation of momentum to find the final velocity of the car/coal system:
p_initial = p_final
315.589 kg·m/s = (m1 + m2) * v_final
Substituting the given masses:
315.589 kg·m/s = (452 kg + 179 kg) * v_final
315.589 kg·m/s = 631 kg * v_final
v_final = 315.589 kg·m/s / 631 kg
v_final ≈ 0.500 m/s
Therefore, the velocity of the car/coal system after the coal has come to rest in the car is approximately 0.500 m/s.