Question

Using dimensional anaysis,shown that the kinetic energy of a body of mass m moving with a velocity v varies as $m{v}^2$

Answer 1

Given that,

Mass $=m$

Velocity $=v$

We know that,

Dimension of energy

$E=\left[M^1{L}^2{T}^{-2}\right]$

Now, mass

$M=\left[M^1{L}^0{T}^0\right]$

Now, velocity

$v=\left[M^0{L}^1{T}^{-1}\right]$

We know that,

Suppose

$E=k{\left[m\right]}^x{\left[v\right]}^y$

Where, k is proportionality constant which is a dimensionless

$\left[M^1{L}^2{T}^{-2}\right]={\left[M^1{L}^0{T}^0\right]}^x{\left[M^0{L}^1{T}^{-1}\right]}^y$

$\left[M^1{L}^2{T}^{-2}\right]=\left[M^x{L}^y{T}^{-y}\right]$

Now, compare the power

$x=1$and$y=2$

So, if $k=\frac{1}{2}$

$E=\frac{1}{2}m{v}^2$

Hence, the kinetic energy is $\frac{1}{2}m{v}^2$