**Kinetic Energy and its Formula**

Kinetic energy (K) is the energy possessed by an object due to its motion. It depends on the mass (m) of the object and its velocity (v). The formula for kinetic energy is given by:

K = ½ mv²

**Determining the Dimensional Formula for Kinetic Energy**

To determine the dimensional formula for kinetic energy (K), we need to express it in terms of the fundamental quantities: velocity (p), acceleration (q), and density (r). Let's break down the formula and determine the dimensions of each term.

K = ½ mv²

Breaking down the terms:

- The mass (m) has the dimension of [M], which represents mass.
- The velocity (v) has the dimension of [LT⁻¹], which represents length divided by time.
- The ½ is a dimensionless constant.

**Combining the Dimensions**

To determine the dimensional formula for kinetic energy, we need to combine the dimensions of mass and velocity. Since kinetic energy is the product of mass and velocity squared, we can express it as:

[K] = [M] × [LT⁻¹]²

Simplifying the dimensions:

[K] = [M] × [L²T⁻²]

Therefore, the dimensional formula for kinetic energy (K) in terms of the given fundamental quantities (p, q, r) is:

[K] = [M] × [L²T⁻²]

**Explanation of the Dimensional Formula**

The dimensional formula for kinetic energy shows that it is composed of the dimensions of mass ([M]), length squared ([L²]), and time squared ([T⁻²]). This indicates that the kinetic energy of an object is directly proportional to its mass and the square of its velocity.

By using the given fundamental quantities (p, q, r) as velocity, acceleration, and density, respectively, we can express the dimensional formula for kinetic energy in terms of these quantities as:

[K] = [M] × [L²T⁻²]

This dimensional formula allows us to analyze and compare kinetic energy in various systems and situations based on the given fundamental quantities.

L^4rt^-1