**Dimensional Analysis**

Dimensional analysis is a technique used in physics to analyze and understand the relationships between physical quantities. It involves studying the dimensions of various fundamental quantities and expressing the dimensions of derived quantities in terms of these fundamental quantities.

In this case, we are given that the fundamental quantities are velocity (v), acceleration (a), and density (ρ). We need to find the dimensional formula for kinetic energy (K).

**Kinetic Energy (K)**

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

K = 1/2 * m * v^2

Where:
- K represents the kinetic energy
- m represents the mass of the object
- v represents the velocity of the object

**Dimensional Formula for Kinetic Energy**

To find the dimensional formula for kinetic energy, we need to analyze the dimensions of the various quantities involved in the formula.

- Velocity (v): The dimensional formula for velocity is [LT^-1], where [L] represents length and [T] represents time.
- Mass (m): The dimensional formula for mass is [M], where [M] represents mass.
- Kinetic Energy (K): We need to determine the dimensional formula for kinetic energy.

By substituting the dimensional formulas of velocity and mass into the formula for kinetic energy, we can determine the dimensional formula for kinetic energy.

K = 1/2 * m * v^2

Substituting the dimensional formulas:

[K] = [M] * ([LT^-1])^2

Simplifying:

[K] = [M] * [L^2] * [T^-2]

Therefore, the dimensional formula for kinetic energy (K) is [M L^2 T^-2].

**Summary**

The dimensional formula for kinetic energy (K) is [M L^2 T^-2], where [M] represents mass, [L] represents length, and [T] represents time. This formula shows the dependence of kinetic energy on mass, length, and time. Dimensional analysis allows us to understand and express the relationships between physical quantities in terms of their fundamental dimensions.

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