Two bodies have masses in the ratio 5 : 1 and kinetic energies in the ration 125 : 9. Calculate the ratio of their velocities.

We know that,

$\text{K.E.} = \dfrac{1}{2} \times m \times v^2 \\[0.5em]$

Let the two bodies be A and B.

Let mass of body A be m_A and body B be m_B. Let velocity of body A be v_A and body B be v_B.

Given,

Ratio of masses:

$\dfrac {m$A}{mB} = \dfrac {5}{1} \\[0.5em]

Ratio of K.E.:

$\dfrac {KE$A}{KEB} = \dfrac {125}{9} \\[1em] \dfrac {KEA}{KEB} = \dfrac{\dfrac{1}{2} \times mA \times vA^2}{\dfrac{1}{2} \times mB \times vB^2} \\[0.5em] \Rightarrow \dfrac {125}{9} = \dfrac{\dfrac{1}{2} \times mA \times vA^2}{\dfrac{1}{2} \times mB \times vB^2} \\[0.5em] \Rightarrow \dfrac {125}{9} = \dfrac{mA}{mB} \times \Big(\dfrac{vA}{vB}\Big)^2 \\[0.5em] \Rightarrow \dfrac {125}{9} = \dfrac{5}{1} \times \Big(\dfrac{vA}{vB}\Big)^2 \\[0.5em] \Rightarrow \Big(\dfrac{vA}{vB}\Big)^2 = \dfrac{125}{9} \times \dfrac{1}{5} \\[0.5em] \Rightarrow \Big(\dfrac{vA}{vB}\Big)^2 = \dfrac{25}{9} \\[0.5em] \Rightarrow \dfrac{vA}{vB} = \sqrt{\dfrac{25}{9}} \\[0.5em] \Rightarrow \dfrac{vA}{vB} = \dfrac{5}{3} \\[0.5em]

∴ Ratio of their velocities = 5:3