91Ó°ÊÓ

When a spring is compressed or stretched from its natural length, it stores energy. This energy is known as elastic potential energy. The formula to calculate this form of potential energy is \[E_e = \frac{1}{2}kx^2\]where \(E_e\) represents the elastic potential energy, \(k\) is the spring constant—a measure of a spring's stiffness—and \(x\) is the displacement from the spring's equilibrium position.
In the context of our toy gun problem, when the spring is compressed by 5.0 cm, the stored elastic potential energy is calculated using \(k = 8.0 \mathrm{~N/m}\) and \(x = 0.05 \mathrm{m}\). This energy is then used to propel the soft rubber sphere once the spring is released.
To comprehend this in everyday terms, think about pushing down on a spring-loaded toy car. The harder you push, the more energy is stored, and the further the car shoots across the room when you let go. The principle at play is the same for our toy gun and bullet.

The work-energy principle is fundamental to understanding how forces acting on an object affect its kinetic energy. The principle states that the work done on an object is equal to the change in its kinetic energy.
In mathematical terms, we express work done as\[W_f = f_fd\]where \(W_f\) is the work done by the frictional force (\(f_f\)) acting over a distance (\(d\)). In our problem, the constant frictional force acts over the length of the barrel to resist the sphere's motion and therefore 'consumes' some of the stored elastic potential energy during the launch.
It's as if you're sliding a box across the floor—the friction between the box and floor resists the motion, requiring you to exert more energy to achieve the same distance. Similarly, the rubber sphere's initial elastic potential energy is reduced by the work done against friction, influencing its final speed as it exits the barrel.

Frictional force is the resistance that one surface or object encounters when moving over another. It is the force that hinders the relative motion of two surfaces in contact. The constant frictional force in our problem is the resistance felt by the sphere as it pushes through the toy gun's barrel.
In this scenario, the frictional force can be thought of as a small hand gripping the sphere as it travels along the barrel, exerting a drag that opposes its forward motion. By doing so, it does work on the sphere proportional to the distance the sphere covers inside the barrel, given by the equation\[W_f = f_fd\]As the sphere moves the 15 cm length of the barrel, the frictional force does work against it, reducing its kinetic energy and final speed as it exits. The greater the frictional force or the longer the barrel, the more energy is lost to friction, and the slower the sphere will emerge.