Question

Two bodies of mass 4 kg and 25 kg have equal kinetic energy. The ratio of their velocities will be?​

Answer 1

Given:

Two bodies of mass 4 kg and 25 kg have equal kinetic energy

To Find:

The ratio of their velocities

Solution:

The relation between the mass of an object (m), its momentum (p), and kinetic energy (KE) is as follows:

$p = \sqrt{2mKE}$.............(a)

Let mass, kinetic energy, and momentum of first object be m₁, KE, p₁ and for second object be m₂, KE, and p₁ respectively.

Putting the values in eq (a), we get

$\frac{p_{1} }{p_{2} } = \frac{\sqrt{2.4.KE} }{\sqrt{2.25.KE} }$

⇒ $\frac{p1}{p2} = \frac{2}{5}$ ...........(b)

Since, Momentum = mass x velocity

⇒ p = mv

⇒$\frac{p1}{p2}= \frac{m1v1}{m2v2}$

⇒$\frac{p1}{p2}= \frac{4.v1}{25v2}$

Putting values of momentum from eq(b)

$\frac{2}{5}= \frac{4v1}{25v2}$

⇒ $\frac{v1}{v2}= \frac{5}{2}$

Hence, the ratio of their velocities is 5:2

Answer 2

Given:

Two bodies of mass 4 kg and 25 kg have equal kinetic energy.

To find:

Ratio of velocity of the objects ?

Calculation:

Let kinetic energy of both the bodies be K :

For 1st body :

$\therefore \: K = \dfrac{1}{2} (m1) {(v1)}^{2}$

$\implies\: {(v1)}^{2} = \dfrac{2K}{m1}$

$\implies\:v1= \sqrt{\dfrac{2K}{m1} }$

For 2nd body:

$\therefore \: K = \dfrac{1}{2} (m2) {(v2)}^{2}$

$\implies\: {(v2)}^{2} = \dfrac{2K}{m2}$

$\implies\:v2= \sqrt{\dfrac{2K}{m2} }$

So, required ratio :

$\therefore \: v1 : v2 = \sqrt{m2} : \sqrt{m1}$

$\implies \: v1 : v2 = \sqrt{25} : \sqrt{4}$

$\implies \: v1 : v2 = 5 : 2$

So, final answer is:

$\boxed{ \bf{\: v1 : v2 = 5 : 2}}$