An open train car, with a mass of 2070 kg, coasts along a horizontal track at the speed 2.31 m/s. The car passes under a loading chute and, as it does so, gravel falls vertically into it for 3.25 s at the rate of 491 kg/s. What is the car's speed after the loading is completed?
To find the car's speed after the loading is completed, we need to apply the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum of a system remains constant if no external forces are acting on it. In this case, the system consists of the open train car and the falling gravel.
First, we need to calculate the initial momentum of the car and the gravel. The initial momentum of the car can be calculated using the formula:
Initial momentum of the car = mass of the car × initial velocity of the car
Given:
Mass of the car = 2070 kg
Initial velocity of the car = 2.31 m/s
Initial momentum of the car = 2070 kg × 2.31 m/s
Next, we need to calculate the momentum of the gravel falling into the car. The momentum of the gravel can be calculated using the formula:
Momentum of the gravel = mass of the gravel × velocity of the gravel
Given:
Mass of the gravel = 491 kg/s × 3.25 s (as the gravel is falling for 3.25 seconds and the rate is given in kg/s)
Momentum of the gravel = (491 kg/s × 3.25 s) × 0 (the velocity of the gravel is assumed to be 0 as it falls vertically)
Since the velocity of the gravel is assumed to be zero, the momentum of the gravel is also zero.
Now, we can calculate the total initial momentum of the system:
Total initial momentum = Initial momentum of the car + Momentum of the gravel
= (2070 kg × 2.31 m/s) + 0
The total initial momentum of the system is equal to the final momentum of the system, as there are no external forces acting on it.
Finally, we can use the principle of conservation of momentum to find the final velocity of the car. Rearranging the formula, we get:
Final velocity of the car = Total initial momentum of the system / Mass of the car
Final velocity of the car = (2070 kg × 2.31 m/s) / 2070 kg
Simplifying the expression, the final velocity of the car is equal to 2.31 m/s.
Therefore, the car's speed after the loading is completed is still 2.31 m/s.