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.

Short answer

Answer: The speed of the train car decreases as it fills with raindrops because the mass of the train car increases due to the raindrops coming to rest inside the train car, and momentum is conserved in the process.

Step-by-step solution

Understand the Conservation of Linear Momentum

Linear momentum is a vector quantity that represents the amount of motion an object has. It is equal to the mass of the object multiplied by its velocity. In an isolated system, the total linear momentum is conserved. This means that the total momentum before the rain starts falling into the train car will be equal to the total momentum after the rain falls into the train car.

Calculate the initial momentum of the train car

The initial momentum of the train car (before rain starts falling) can be calculated using this formula: Initial momentum = mass of train car × initial velocity. Let the mass of the train car be \(m_{car}\) and its initial velocity be \(v_{0}\). Then, Initial momentum = \(m_{car} \times v_{0}\)

Calculate the momentum of raindrops falling into the train car

As the rain falls into the train car, the mass of the train car increases, but the raindrops come to rest inside the car. Let the mass of the raindrops be \(\Delta m_{rain}\), and their initial velocity be \(0\), as they are considered to be at rest inside the train car after they fall into it. The momentum of raindrops inside the train car is \(\Delta m_{rain} \times 0\), which is \(0\).

Calculate the momentum of the combined system

The combined system consists of the train car with the increased mass (due to the rain). Let the final velocity of the combined system be \(v_{f}\). So, the momentum of the combined system is: Momentum of combined system = (mass of train car + mass of raindrops) × final velocity Momentum of combined system = \((m_{car} + \Delta m_{rain}) \times v_{f}\)

Apply the conservation of linear momentum

Since the total momentum is conserved, the initial momentum of the train car should be equal to the momentum of the combined system after the rain falls into the train car: \(m_{car} \times v_{0} = (m_{car} + \Delta m_{rain}) \times v_{f}\) Now, let's solve for the final velocity of the train car (combined system), \(v_{f}\): \(v_{f} = \frac{m_{car} \times v_{0}}{m_{car} + \Delta m_{rain}}\) We can notice that as the mass of the raindrops (\(\Delta m_{rain}\)) increases, the denominator becomes larger. Therefore, the final velocity (\(v_{f}\)) of the combined system (train car + rain) decreases.

Conclusion

Based on the principle of conservation of linear momentum, we can conclude that the speed of the train car decreases as it fills with rain. This occurs because the mass of the train car increases due to the raindrops, they come to rest inside the train car, and momentum is conserved in the process.

Key concepts

Momentum Conservation Principle

Imagine you're watching a spectacular ice-skating performance where a skater throws a snowball while spinning. Regardless of the snowball's departure, the skater continues to spin without abrupt changes. This mesmerizing act mirrors the momentum conservation principle, a cornerstone in the realms of physics.

This principle asserts that in an isolated system—free from external forces—the total momentum remains unaltered over time. An isolated system means nothing enters or leaves the system, and no external forces are acting on it, much like the initially engineless train car moving on a frictionless track.

In essence, the momentum—a product of mass and velocity—before any interaction must be equal to the total momentum after the interaction. This concept is pivotal when predicting the outcomes of collisions or, as with our scenario, when rain joins its journey with a moving train car.

Initial Momentum Calculation

The show starts with the train car gliding alone on the track—this scene sets the stage for initial momentum calculation. It's the opening act, representing the momentum of the system before the raindrop performance begins.

The initial momentum is a simple calculation: take the mass of the train car and multiply it by its speed (\( m_{car} \times v_{0} \)). This arithmetic dance gives us a value, the system's momentum, before the rain makes its entrance. Here, the simplicity is key, as understanding the initial state is essential to predict how the system will evolve.

Combined System Momentum

The plot thickens as rain enters the stage, merging with our lonely train car. They form what we call a combined system, a new ensemble which includes the mass of our original car and the newly arrived raindrops. But what happens to their momentum now?

Since momentum must be conserved, we calculate the momentum of the combined system using the formula \[ (m_{car} + \Delta m_{rain}) \times v_{f} \. The final act reveals that although their dance is a duo, the combined system's final velocity (\( v_{f} \) slows down due to the increased total mass, highlighting the harmony between mass increase and velocity in maintaining momentum conservation.

Momentum of Raindrops

In our train car saga, imagine the raindrops as individual performers dropping onto the stage, each carrying their own momentum. Initially, as they descend vertically, their horizontal momentum is zero—this might seem like a trivial detail but is actually pivotal in the grand scheme of momentum management.

When the raindrops join the train car, they relinquish their free-fall performance and adopt the car's horizontal motion, effectively bringing their momentum to \(0\) in the context of the system's horizontal movement. This act is a crucial piece of the puzzle as it underlines why the car's original momentum, uninterrupted by the entrance of the raindrops, dictates the final velocity of the newly formed combined system.