A mine car (mass 380 kg) rolls at a speed of 0.5 m/s on a horizontal track, as the drawing shows. A 150 kg chunk of coal has a speed of 0.80 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 determine the velocity of the car/coal system after the coal has come to rest in the car, we need to apply the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of an isolated system remains constant if no external forces act on it. In this case, we can consider the car/coal system as an isolated system.
We can calculate the initial momentum and the final momentum of the system and equate them to find the final velocity.
1. Calculate the initial momentum: Momentum (p) is given by the product of mass (m) and velocity (v). The initial momentum of the car can be calculated as follows:
Initial momentum of car = Mass of car × Velocity of car
= 380 kg × 0.5 m/s
= 190 kg·m/s
The initial momentum of the chunk of coal can be calculated as follows:
Initial momentum of coal = Mass of coal × Velocity of coal
= 150 kg × 0.80 m/s
= 120 kg·m/s
2. Calculate the final momentum: After the coal has come to rest and is inside the car, the final momentum of the car/coal system is the sum of the mass of the car and the mass of the coal combined, multiplied by the final velocity of the system (v):
Final momentum of car/coal system = (Mass of car + Mass of coal) × Velocity of system
= (380 kg + 150 kg) × v
Since the coal has come to rest, its final velocity is 0 m/s. Therefore, we have:
Final momentum of car/coal system = (380 kg + 150 kg) × 0
= 0 kg·m/s
3. Equate the initial and final momenta: According to the conservation of momentum, the initial momentum of the system must be equal to the final momentum:
Initial momentum of car + Initial momentum of coal = Final momentum of car/coal system
190 kg·m/s + 120 kg·m/s = 0 kg·m/s
4. Solve for the final velocity: Rearranging the equation, we can solve for the final velocity:
(380 kg + 150 kg) × v = 0 kg·m/s
530 kg × v = 0 kg·m/s
v = 0 kg·m/s / 530 kg
v ≈ 0 m/s
Therefore, the velocity of the car/coal system after the coal has come to rest in the car is approximately 0 m/s, indicating that the system is at rest.
momentum is conserved. Final momentum is masscoal*.8 +minecarmass*.5=(totalmass)v
solve for v