Problem 19 - University Physics with Modern Physics

Question

An open train car moves with speed \(v_{0}\) on a flat frictionless railroad track, with no engine pulling it. It begins to rain. The rain falls straight down and begins to fill the train car. Does the speed of the car decrease, increase, or stay the same? Explain.

Step-by-Step Solution

Answer

Answer: The speed of the open train car decreases as rain falls into it, due to the conservation of momentum and the increase in the train car's mass.

Step 1: 1. Understand the concept of conservation of momentum

According to the law of conservation of momentum, the total momentum of a closed system (no external forces acting on it) remains constant if no net external forces act upon it. The momentum of an object is given by the product of its mass and velocity (p = mv). In this problem, we will analyze the momentum conservation of the train car and the falling rain.

Step 2: 2. Determine the initial momentum of the system

Before the rain starts falling, the train car has a momentum of p_initial = m_car * v0, where m_car is the mass of the car and v0 is its initial speed. The rain itself initially has no horizontal momentum, as it falls straight down from the sky.

Step 3: 3. Analyze the momentum when rain falls in the train car

As the rain falls into the train car, it also starts moving horizontally with the speed of the car since there's no friction to stop it. When a certain mass (m_rain) of rain falls in the train car, the train car's mass increases to m_car + m_rain.

Step 4: 4. Establish the momentum conservation equation

Since there are no external horizontal forces acting on the system, the total initial momentum should equal the final momentum. Therefore, we can set up the following equation: p_initial = p_final m_car * v0 = (m_car + m_rain) * v_final

Step 5: 5. Solve for the final speed (v_final) of the train car

Rearrange the equation to solve for v_final: v_final = (m_car * v0) / (m_car + m_rain) Since m_car + m_rain is always greater than m_car (as m_rain > 0), we can see that v_final will always be less than v0.

Step 6: Conclusion

The speed of the car decreases as the rain falls into the train car and increases its mass. This can be explained by the conservation of momentum, as the increasing mass of the system results in a lower speed for the train car to maintain the same total momentum.