91Ó°ÊÓ

Kinetic energy is a fundamental concept in physics and refers to the energy an object possesses due to its motion. It is mathematically expressed as \( KE = \frac{1}{2} mv^2 \), where \( m \) is the mass of the object and \( v \) is its velocity.
In the context of the exercise, the block starts from rest, meaning its initial kinetic energy is zero. As the block is accelerated by the spring, its velocity increases until it reaches a maximum value when it leaves the spring. This point is when the block has its maximum kinetic energy.
The principle of conservation of energy tells us that this maximum kinetic energy is equal to the energy initially stored in the compressed spring. Once the block leaves the spring and moves across the floor, the kinetic energy is gradually converted into thermal energy due to friction until the block comes to a stop. Essentially, the kinetic energy at maximum is transformed into work done against the frictional force.

Frictional force is the resistive force that opposes the motion of an object when it slides over a surface. In this exercise, it is crucial because it stops the block by transforming kinetic energy into thermal energy.
To calculate the frictional force \( f_k \), we use the formula \( f_k = \mu_k \, m \, g \), where \( \mu_k \) is the coefficient of kinetic friction, \( m \) is the mass of the block, and \( g \) is the acceleration due to gravity. This force is responsible for the conversion of the block's kinetic energy into heat as it moves over the floor.
The relationship between the frictional force and the work done can be expressed as the work-energy principle, which states that the work done by friction over a distance \( D \) is equal to the decrease in the block's kinetic energy. The calculated frictional force times the distance gives the total work done by friction, leading to the increase in the thermal energy of the block-floor system.

Elastic potential energy is energy stored in elastic materials as a result of their stretching or compression. In the problem, this energy is initially stored in the compressed spring and is what accelerates the block from rest.
To find the elastic potential energy, we use the formula \( U = \frac{1}{2} k x^2 \), where \( k \) is the spring constant and \( x \) is the displacement from the spring's relaxed length. This potential energy is entirely converted into kinetic energy as the spring returns to its relaxed length, propelling the block forward.
Solving for the compression distance is done by equating the kinetic energy at maximum, \( K_{max} \), which is observed when the block is at the spring's relaxed length, to the initial elastic potential energy. Through rearranging and solving the formula, we determine the original compression length of the spring, helping understand the spring's role in the motion of the block.