Question

Two bodies of masses ${}^{\prime }{m}^{\prime }$ and ${}^{\prime }4m^{\prime }$ are having linear momentum in the ratio of $2:1$, respectively. The ratio of their kinetic energy is

• A. $4:1$
• B. $1:4$
• C. $1:8$
• D. $16:1$

Answer 1

The correct option is D $16:1$

Lets assume that
$P_1=$ Linear momentum of mass ${}^{\prime }{m}^{\prime }$
$K{E}_1=$ Kinetic energy of mass ${}^{\prime }{m}^{\prime }$
$P_2=$ Linear momentum of mass ${}^{\prime }4m^{\prime }$
$K{E}_2=$ Kinetic energy of mass ${}^{\prime }4m^{\prime }$

We also know that, the relation between kinetic energy and linear momentum is given by,
$KE=\frac{P^2}{2m}$
$\frac{K{E}_1}{K{E}_2}=\frac{\frac{P_1^2}{2m}}{\frac{P_2^2}{2\times 4m}}$

$\because \frac{P_1}{P_2}=\left(\frac{2}{1}\right)$ (given)

$\therefore \frac{K{E}_1}{K{E}_2}={\left(\frac{2}{1}\right)}^2\times 4=(16:1)$

Hence option D is the correct answer.