In a railroad switchyard, a 56 -ton freight car is sent at \(7.0 \mathrm{mi} / \mathrm{h}\) toward a 31 -ton car moving in the same direction at \(2.6 \mathrm{mi} / \mathrm{h} .\) (a) What's the speed of the cars after they couple? (b) What fraction of the initial kinetic energy was lost in the collision?

Kinetic energy is a form of energy that objects possess due to their motion. It's quantifiable as \( KE = \frac{1}{2}mv^2 \) where \( m \) is the mass of the object and \( v \) is its velocity. This formula illustrates that kinetic energy depends directly on the mass of an object and exponentially on its speed; even a small increase in speed will produce a significant rise in kinetic energy.

For instance, in the given exercise, the initial kinetic energies of the 56-ton and 31-ton freight cars are computed separately before they collide. The sum of these individual kinetic energies represents the total kinetic energy of the system before the collision. Kinetic energy is crucial to the concept of work and energy in physics, informing us how much work the moving freight cars can potentially do, based on their masses and velocities, before colliding.

An inelastic collision is one in which the colliding objects stick together after impact, which is exactly what happens when railroad cars couple. Unlike elastic collisions, where kinetic energy is conserved, inelastic collisions do not conserve kinetic energy even though momentum is conserved. Some of the kinetic energy is converted to other forms of energy, such as heat, sound, or deformation.

In the context of the freight cars in the exercise, once they couple and move together, they create one single body with a combined mass and a common velocity. With kinetic energy loss inevitable, the calculations should reveal a decrease in total kinetic energy post-collision, which is expressed as a fractional loss relative to the pre-collision kinetic energy.

Momentum conservation is a fundamental principle in physics that states the total momentum of a closed system remains constant if no external forces act upon it. Momentum, represented by \( p \) and calculated as the product of mass \( m \) and velocity \( v \) (\( p = mv \) ), is a vector quantity, meaning it has both magnitude and direction. During any collision, the direction and magnitude of momentum play crucial roles.

In the scenario with the coupling freight cars, momentum conservation allows us to calculate the velocity of the cars post-collision. Since we're dealing with a closed system with no external forces (ignoring friction and air resistance for simplicity), the sum of the products of mass and velocity for both cars before the collision must equal the product of the combined mass and their common velocity after the collision. This principle is at the heart of solving part (a) of the exercise, where we find the common speed of the coupled cars. Understanding momentum conservation is vital for not only this exercise but for analyzing any collision in physics.