A mine car, mass= 440 kg rolls at a speed of 0.50 m/s on a horizontal track. A 150-kg chunk of coal has a speed of 0.80 m/s when it leaves the chute. The chunk of coal falls into the car at a 25 degree angle. Determine the velocity of the car/coal system after the coal has come to rest in the car.
The answer depends upon whether the 25 degree angle is tilted in the direction of motion of the car or in the reverse direction. In any case, apply conservation of horizontal momentum
Thank you very much
To determine the velocity of the car/coal system after the coal has come to rest in the car, we can use the principle of conservation of linear momentum.
The momentum of an object is given by the product of its mass and velocity.
Before the coal falls into the car, the momentum of the car is given by:
Momentum_car = mass_car * velocity_car
After the coal falls into the car, the momentum of the car/coal system is equal to the sum of the momentum of the car and the momentum of the coal.
Let's calculate the momentum of the car and the momentum of the coal separately.
1. Momentum of the car:
Momentum_car = mass_car * velocity_car
= 440 kg * 0.50 m/s
= 220 kg.m/s
2. Momentum of the coal:
To calculate the momentum of the coal, we need to find its final velocity after it comes to rest in the car.
The velocity of the coal before it falls into the car can be divided into its horizontal and vertical components. The horizontal component of velocity remains the same after it falls into the car, while the vertical component reduces to zero when it comes to rest.
Horizontal component of velocity = velocity_coal * cos(theta)
= 0.80 m/s * cos(25°)
Vertical component of velocity = velocity_coal * sin(theta)
= 0.80 m/s * sin(25°)
The momentum of the coal before it falls into the car is given by:
Momentum_coal_before = mass_coal * velocity_coal_before
= 150 kg * (0.80 m/s * cos(25°) + 0.80 m/s * sin(25°))
Since the coal comes to rest in the car, its final velocity is zero. Thus, the momentum of the coal after it comes to rest is zero.
Now, the momentum of the car/coal system after the coal comes to rest in the car is given by:
Momentum_car_system = Momentum_car + Momentum_coal
= 220 kg.m/s + 0
Therefore, the velocity of the car/coal system after the coal has come to rest in the car is zero.
To determine the final velocity of the car/coal system after the coal has come to rest in the car, we need to use the principle of conservation of momentum.
The principle of conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, as long as no external forces are acting on the system.
In this case, the initial momentum of the car and the coal is the sum of their individual momenta. The final momentum of the car/coal system after the coal comes to rest in the car is equal to the final momentum of the car alone since the coal has zero velocity.
Let's break down the steps to find the final velocity of the car/coal system:
Step 1: Calculate the initial momentum of the car and the coal separately.
The initial momentum of the car is given by:
momentum_car_initial = mass_car * velocity_car_initial
Given:
mass_car = 440 kg
velocity_car_initial = 0.50 m/s
So, momentum_car_initial = 440 kg * 0.50 m/s
The initial momentum of the coal is given by:
momentum_coal_initial = mass_coal * velocity_coal_initial
Given:
mass_coal = 150 kg
velocity_coal_initial = 0.80 m/s
So, momentum_coal_initial = 150 kg * 0.80 m/s
Step 2: Calculate the final momentum of the car/coal system.
Since the coal comes to rest in the car, its final momentum is zero.
So, the final momentum of the car/coal system is equal to the final momentum of the car.
Therefore, momentum_car_final = final momentum of the car/coal system = 0
Step 3: Equate the initial and final momenta and solve for the velocity of the car/coal system.
momentum_car_initial = momentum_car_final
440 kg * 0.50 m/s = 0
Simplifying the equation:
220 kg * velocity_car_final = 0
Therefore, the final velocity of the car/coal system is 0 m/s.
So, after the coal comes to rest in the car, the car/coal system will have a velocity of 0 m/s.