Current automotive standards call for bumpers that sustain essentially no damage in a \(4-\mathrm{km} / \mathrm{h}\) collision with a stationary object. As an automotive engineer, you'd like to improve on that. You've developed a spring-mounted bumper with effective spring constant \(1.3 \mathrm{MN} / \mathrm{m} .\) The springs can compress up to \(5.0 \mathrm{~cm}\) before damage occurs. For a \(1400-\mathrm{kg}\) car, what do you claim as the maximum collision speed?

When a moving object comes to a stop, its kinetic energy must go somewhere. This is a classic illustration of energy transformation. In physics, kinetic energy is the energy an object has due to its motion and is given by the formula \( \frac{1}{2} mv^{2} \), where \( m \) is the mass and \( v \) is the velocity of the moving object.
In our case, a car's kinetic energy during a collision is absorbed by the bumper's springs and becomes potential energy, which is the energy stored within the springs due to their compression. The potential energy in a spring is calculated by the formula \( \frac{1}{2} kx^{2} \), where \( k \) is the spring constant expressing the spring's stiffness, and \( x \) is the compression distance.

This transformation occurs because energy in a closed system is conserved. So, when a car hits a spring-mounted bumper, its kinetic energy decreases down to zero as the energy is transferred into the springs, compressing them. If the springs are not compressed to the point of damage, this process is reversible when the springs return to their original shape, though some energy might be lost to other forms, such as heat or sound.

Understanding how springs behave under force is crucial in scenarios where cushioning impact is necessary, like in car bumpers. This is where Hooke's Law comes to play. Hooke's Law states that the force needed to compress or extend a spring by some distance \( x \) is proportional to that distance. This is mathematically expressed as \( F = -kx \), where \( F \) is the force applied to the spring, \( k \) is the spring constant, and \( x \) is the displacement from the spring's equilibrium position.

The negative sign indicates that the force applied by the spring is in the opposite direction of the applied force — this is a restoring force aiming to bring the spring back to its relaxed state. Hooke’s Law is fundamental in designing spring-mounted systems because it allows engineers to predict how a spring will respond to a given force, ensuring that the car’s bumper will effectively absorb the impact without causing damage.

The principle of energy conservation is a cornerstone of physics, particularly when analyzing collisions. It asserts that in a closed system, energy cannot be created or destroyed, only transformed from one form to another or transferred from one object to another. In the context of an automobile collision with a spring-mounted bumper, this principle ensures that the kinetic energy of the moving vehicle is not lost but is instead changed into potential energy as the bumper's springs compress.

During such an elastic collision where no energy is dissipated as heat or sound, the total mechanical energy (kinetic plus potential) before and after the collision should remain the same. By calculating and equating the kinetic energy of the car before the crash and the potential energy of the compressed springs, engineers can determine essential parameters like the maximum collision speed that can be withstood without damage, as was shown in the step-by-step solution of the given exercise.