The Kinetic energy (n) of a body is found to depend on mass (x) and Velocity (y) of a body. Use the method of dimension to show the Relation between 'n', 'x' and 'y'

The kinetic energy is denoted by $n$. Its dimensions are:

$$[n] = [M L^2 T^{-2}]$$

The mass is denoted by $x$. Its dimensions are:

$$[x] = [M]$$

The velocity is denoted by $y$. Its dimensions are:

$$[y] = [L T^{-1}]$$

Let's assume that kinetic energy $n$ depends on mass $x$ and velocity $y$ in the following form, where $k$ is a dimensionless constant and $a, b$ are exponents to be determined:

$$n = k \cdot x^a y^b$$

Equating the dimensions on both sides of the equation:

$$[M L^2 T^{-2}] = [M]^a [L T^{-1}]^b$$ $$[M L^2 T^{-2}] = [M^a L^b T^{-b}]$$

Comparing the powers of M:

$$1 = a$$

Comparing the powers of L:

$$2 = b$$

Comparing the powers of T:

$$-2 = -b \implies b = 2$$

We found $a=1$ and $b=2$. Substituting these values:

$$n = k \cdot x^1 y^2$$ $$n = k \cdot x y^2$$

The method of dimensions determines the proportionality of the physical quantities. The dimensionless constant $k$ cannot be determined by this method, but from physics, we know $k = \frac{1}{2}$.

The relation between $n$, $x$, and $y$ is:

$$n \propto x y^2$$