How do I solve this?
A mine car (mass=380 kg) rolls at a speed of 0.50 m/s on a horizontal track, as the drawing shows. A 170-kg chunk of coal has a speed of 0.92 m/s when it leaves the chute (at a 25 degree incline). Determine the speed of the car–coal system after the coal has come to rest in the car.
To solve this problem, we need to use the principle of conservation of momentum, which states that the total momentum before an event is equal to the total momentum after the event (assuming no external forces are acting on the system).
First, let's determine the initial momentum of the system before the coal enters the car. The momentum of an object is given by the product of its mass and velocity. We can calculate the initial momentum of the car and the coal separately:
Initial momentum of the car (Pcar) = mass of the car (mcar) × velocity of the car (vcar)
Pcar = 380 kg × 0.50 m/s
Initial momentum of the coal (Pcoal) = mass of the coal (mcoal) × velocity of the coal (vcoal)
Pcoal = 170 kg × 0.92 m/s
Next, we need to calculate the final momentum of the system when the coal comes to rest in the car. Since the coal has come to rest, its final velocity is zero. The final momentum of the system will be the combined momentum of the car and the coal:
Final momentum of the car-coal system (Pfinal) = Pcar + Pcoal
Using the calculated values for Pcar and Pcoal, we can find Pfinal.
Finally, we can find the speed of the car-coal system after the coal has come to rest. The speed is the magnitude of the final velocity and is given by dividing the final momentum by the total mass of the car and the coal:
Speed of the car-coal system (vfinal) = Pfinal / (mcar + mcoal)
Substituting the calculated value for Pfinal and the given masses of the car and the coal, we can solve for vfinal.