Question

Derive the formula for kinetic energy of
a particle having mass m and velocity v
using dimensional analysis​

Answer 1

Answer:-

To Prove:-

$\bf \longrightarrow k = \frac{1}{2} mv {}^{2} .$

• Using Dimensional Formula.

So let,

• Kinetic energy be k.

• Mass be m.

• velocity be v.

Now,

We know that,

• Kinetic energy depends upon mass and velocity.

So Let,

$\\ \tt\mapsto k \propto m {}^{a} \: \: \: \: also \: \: \: \: k \propto {m}^{b}.$

$\\ \tt\mapsto k \propto m {}^{a}v {}^{b}.$

$\\ \large\mapsto \boxed{ \bf k = cm {}^{a} {v}^{b} }....(1)$

Where c is constant.

Now,

By writing Dimensions formula both sides we get,

$\\ \tt\mapsto [ML^2T^{-2}] = M^a[LT^{-1}]^b.$

$\\ \bf\mapsto [ML^2T^{-1}] = M^aL^bT^{-b}.$

On Comparing Both Sides:-

$\\ \tt\mapsto M^1 = M^a.$

$\\ \large \bf\mapsto a = 1.$

And,

$\\ \tt\mapsto L^2 = L^b.$

$\\ \large\tt\mapsto b = 2.$

So we get the values of a and which are 1 and 2 respectively.

Therefore,

By Putting the values of a and b in eq(1) we get,

$\\ \mapsto \bf k = cm {}^{a} {v}^{b}.$

$\\ \tt\mapsto k =c m {}^{1} {v}^{2} .$

$\\ \tt\mapsto k = cmv {}^{2}.$

Also, in kinetic energy value of constant Is 1/2 therefore value of c is 1/2.

$\\ \large \red{\mapsto \boxed{ \blue{ \bf k = \frac{1}{2} mv {}^{2} .}}}$

...Hence Proved...

(See the attachment to know all the steps of how to prove any formula using dimensional analysis).