The Physics of Sand Flow and Freight Car Motion

Introduction:
In this scenario, we have a freight car of mass M that contains a mass of sand m. At t=0, a constant horizontal force F is applied to the freight car, and at the same time, a port in the bottom is opened to let the sand flow out at a constant rate dm/dt. We need to determine the speed of the freight car when all the sand is gone.

Key Concepts:
To solve this problem, we will apply the principle of conservation of momentum and the equation of motion for the freight car. Here are the key concepts we will use:

1. Conservation of Momentum: The total momentum of an isolated system remains constant if no external forces act on it. In this case, the system consists of the freight car and the sand.

2. Equation of Motion: The equation of motion relates the net force acting on an object to its mass and acceleration. In this case, the net force acting on the freight car is the applied force F minus the force exerted by the sand flowing out.

Steps to Solve:
1. Determine the rate of sand flow: dm/dt represents the rate at which the mass of sand is decreasing with time. Since the sand is flowing out at a constant rate, dm/dt is a constant.

2. Calculate the force exerted by the sand: The force exerted by the sand is equal to the rate of sand flow multiplied by the acceleration due to gravity (F_sand = dm/dt * g).

3. Apply the principle of conservation of momentum: Initially, the sand and the freight car are at rest, so the total momentum is zero. When the sand is gone, the final momentum is zero as well. Therefore, the change in momentum of the system is zero.

4. Write the equation of motion for the freight car: The net force acting on the freight car is the applied force F minus the force exerted by the sand. According to Newton's second law, F_net = M * a, where a is the acceleration of the freight car.

5. Solve the equation of motion: Equate the net force to the mass times acceleration and solve for the acceleration.

6. Calculate the final velocity: Once the acceleration is known, use the equation v = u + at to calculate the final velocity of the freight car when all the sand is gone.

Conclusion:
By applying the principle of conservation of momentum and the equation of motion, we can determine the speed of the freight car when all the sand is gone. The key steps involve calculating the force exerted by the sand, applying the principle of conservation of momentum, and solving the equation of motion for the freight car.