Question

Two bodies a and b have mass in ratio 5:1 and kinetic energy in ratio 125:9 find thier ratio of velocity.

Answer 1

Let mass of body A is $m_A$

mass of body B is $m_B$

a/c to question,

$\frac{m_A}{m_B}=\frac{5}{1}$.....(1)

$\frac{K.E_A}{K.E_B}=\frac{125}{9}$.

.......(2)

we know, formula of kinetic energy , $K.E=\frac{1}{2}mv^2$

$\frac{K.E_A}{K.E_B}=\frac{\frac{1}{2}m_Av_A^2}{\frac{1}{2}m_Bv_B^2}$

or, $\frac{K.E_A}{K.E_B}=\frac{m_Av_A^2}{m_Bv_B^2}$

or, $\frac{125}{9}=\frac{m_A}{m_B}\frac{v_A^2}{v_B^2}$

or, $\frac{125}{9}=\frac{5}{1}\left(\frac{v_A}{v_B}\right)^2$

or, $\frac{25}{9}=\left(\frac{5}{3}\right)^2=\left(\frac{v_A}{v_B}\right)^2$

or, $\frac{v_A}{v_B}=\frac{5}{3}$

hence, ratio of their velocities is 5 : 3

Answer 2

Guven,

Ratio of the mass of the body = 5 : 1`

Ratio of the Kinetic Energy of the body = 125/9

Since, K.E. = 1/2 mv²

Therefore, Ratio of K.E.₁ : K.E.₂ = m₁v₁²/m₂v₂²

∴ v₁²/v₂² = K.E.₁/K.E.₂ × m₂/m₁

∴ v₁²/v₂² = 125/9 × 1/5

∴ v₁²/v₂² = 25/9

∴ v₁/v₂ = 5/3

Hence, the ratio of the velocity of the body a to the velocity of the body b is 5 : 3.

Hope it helps.