Conservation of Momentum in a Railroad Car Collision: A Mathematical Analysis

In this article, we will delve into the principles of conservation of momentum and apply them to solve a specific problem involving a collision between two railroad cars. The problem at hand is mathematically intriguing and provides a practical example of applying physics principles in real-world scenarios.

Problem Statement

A 7500-kg railroad car traveling at 18 m/s strikes a second railroad car which is initially at rest. After the collision, both cars stick together and move off with a final speed of 5.0 m/s. The question is: What is the mass of the second car?

Concept of Conservation of Momentum

Conservation of momentum is a fundamental principle in physics. It states that in a closed system, the total momentum remains constant. Mathematically, this can be expressed as:

Pinitial Pfinal

Where P is the total momentum of the system.

Applying the Concept to the Problem

Let's denote the mass of the second car as m2. According to the conservation of momentum:

(7500 kg)(18 m/s) (0 kg)(0 m/s) (7500 kg m2) (5 m/s)

Simplifying this equation:

135000 kg·m/s (7500 kg m2) (5 m/s)

135000 kg·m/s 37500 kg·m/s 5m2·m/s

Subtracting 37500 kg·m/s from both sides:

97500 kg·m/s 5m2·m/s

Dividing both sides by 5:

m2 19500 kg

Alternative Approaches and Considerations

It's important to note that while the primary approach using conservation of momentum provides a direct solution, one cannot simply take the mass of the stationary car as 7500 kg, as it is explicitly stated in the problem. Therefore, the mass of the second car must be 19500 kg for the equations to balance.

Additionally, the problem involves the collision of the cars, which does not conserve kinetic energy. Some energy is lost due to deformation and possibly breaking of pieces, leading to a decrease in the total energy of the system. This highlights the difference between momentum and energy conservation.

A Ballistic Pendulum Analogy

The problem at hand can be compared to a ballistic pendulum, where a mass strikes a free-hanging mass and sticks to it. The conservation of linear Newtonian momentum in such scenarios can be similarly applied to find the unknown mass. In both cases, the initial momentum equals the final momentum, allowing us to solve for the unknown mass.

To summarize, the mass of the second railroad car is 19500 kg. The collision does not conserve kinetic energy due to energy loss from heat and deformation, yet the principle of conservation of momentum remains valid for determining the composite mass and final velocity post-collision.

Understanding and applying these principles is crucial in various fields, from engineering to sports and everyday life. By keeping these fundamental concepts in mind, we can better analyze and solve real-world problems.