In a railroad yard, a \(35,000-\mathrm{kg}\) boxcar moving at \(7.5 \mathrm{m} / \mathrm{s}\) is stopped by a spring-loaded bumper mounted at the end of the level track. If \(k=2.8 \mathrm{MN} / \mathrm{m},\) how far does the spring compress in stopping the boxcar?

Kinetic energy represents the energy that an object possesses due to its motion. It is calculated with the formula: \[ KE = \frac{1}{2}mv^2 \] where \(m\) is the mass of the object and \(v\) its velocity. In the context of the railroad yard problem, a moving boxcar has kinetic energy because of its speed. Imagine this energy as the boxcar's ability to do work, such as pushing against a spring or moving another object. Kinetic energy is a form of mechanical energy that depends not only on the speed of the object but also on its mass. The greater the mass and velocity of the object, the higher its kinetic energy will be.

Potential energy, on the other hand, is the stored energy in an object due to its position or arrangement. In the case of springs, the potential energy can be expressed as:\[ PE = \frac{1}{2}kx^2 \] Here, \(k\) stands for the spring constant, which measures the spring's stiffness, and \(x\) represents the displacement or compression of the spring from its rest position. When the boxcar in our exercise compresses the spring, it stores energy in the spring due to its position. This energy could be released if the spring were allowed to rebound, propelling the boxcar back in the opposite direction with a force proportional to the amount of compression.

The spring constant, denoted by \(k\), is a unique value for each spring that quantifies its stiffness. A higher spring constant means a stiffer spring that requires more force to compress or stretch. The unit of the spring constant is Newtons per meter (N/m). In our exercise, the spring's stiffness is very high at 2.8 MN/m or 2800000 N/m. This indicates that the spring will resist compression with a significant force. The formula to relate the force \(F\) to the spring's displacement \(x\) is given by Hooke's Law, \( F = kx \). When dealing with energy, the spring constant is central in determining how much potential energy is stored in the spring for a given amount of compression.

The principle of energy conservation is fundamental in physics and asserts that energy cannot be created or destroyed in an isolated system, only transformed from one form to another. In the scenario with the boxcar and spring, this principle guarantees that the kinetic energy of the moving boxcar is transformed entirely into the potential energy of the compressed spring (assuming no other energy losses such as friction or air resistance). Thus, by equating the initial kinetic energy to the final potential energy of the spring, \( \frac{1}{2}mv^2 = \frac{1}{2}kx^2 \), we can solve for \(x\), the compression distance. Energy conservation allows us to predict the outcome of many physical situations by simply tracking how energy changes form and is transferred between objects.