91Ó°ÊÓ

Kinetic energy is the energy an object possesses due to its motion. In the context of the exercise, the mass starts from rest and then gains speed as the spring pushes it. This gain in speed translates to an increase in kinetic energy.

The kinetic energy of an object is given by the formula:

• \( KE = \frac{1}{2} m v^2 \)

where:

• \( KE \) is the kinetic energy
• \( m \) is the mass of the object
• \( v \) is the velocity (or speed) of the object

In our problem, the point at which the spring has completely transferred its stored potential energy to the mass is when the mass achieves its maximum speed, thus converting all the potential energy to kinetic energy.

By setting the stored potential energy equal to the kinetic energy formula, we can solve for the mass's greatest speed. This transformation exemplifies the conservation of energy principle, where no energy is lost, just transformed from one type to another. The understanding of kinetic energy and its conversion from potential energy is crucial in explaining motion dynamics.

Hooke's Law describes the behavior of springs and how they exert force when compressed or stretched. It is a fundamental principle explaining the linear relationship between the displacement of a spring and the force it exerts. This principle is very helpful for calculations involving springs, such as the one described in the exercise.

The law is expressed as:

• \( F = -kx \)

where:

• \( F \) represents the restoring force exerted by the spring
• \( k \) is the spring constant, a measure of the spring's stiffness
• \( x \) is the displacement from the spring's equilibrium position

In this exercise, we find the maximum force exerted by using this law at the point of maximum compression or extension. This is useful to determine the maximum acceleration of the mass once the spring is released. By knowing the spring constant and displacement, we can precisely calculate this force, which plays a critical role in understanding how the spring affects the object's motion.

Potential energy in this scenario is stored energy in the compressed spring. When you compress a spring, you are doing work on it, which is stored as elastic potential energy.

The formula for the potential energy stored in a spring is:

• \( PE = \frac{1}{2} k x^2 \)

where:

• \( PE \) is the potential energy
• \( k \) is the spring constant, indicating how stiff the spring is
• \( x \) is the displacement from its rest position

The problem specifies that there is 11.5 Joules of potential energy when the spring is compressed. This energy becomes crucial as it provides the necessary force to propel the mass once released. It showcases how potential energy can be converted into kinetic energy during its motion, reinforcing the conservation of energy principle. Understanding potential energy is vital to predicting and explaining the behavior of the spring system, especially when calculating maximum speeds and accelerations involved.