Question

Two bodies, A and B, have equal kinetic energies. The mass of A is nine times that of B. The ratio of the momentum of A to that of B is:

Answer 1

Given:

Two bodies, A and B, have equal kinetic energies. The mass of A is nine times that of B.

To find:

The ratio of the momentum of A to that of B is ?

Calculation:

Momentum expressed in terms of KE and mass:

$\therefore \: KE = \dfrac{ {p}^{2} }{2m}$

$\implies \: {p}^{2} = 2m(KE)$

$\implies \: p = \sqrt{2m(KE)}$

When KE is constant :

$\boxed{ \implies \: p \propto \sqrt{m}}$

So, taking corresponding ratios:

$\therefore \: \dfrac{p_{A}}{p_{B}} = \sqrt{ \dfrac{ m_{A} }{ m_{B}} }$

$\implies \: \dfrac{p_{A}}{p_{B}} = \sqrt{ \dfrac{ 9 \times m_{B} }{ m_{B}} }$

$\implies \: \dfrac{p_{A}}{p_{B}} = \sqrt{9}$

$\implies \: \dfrac{p_{A}}{p_{B}} = 3$

$\implies \: p_{A} : p_{B} = 3 : 1$

So, required ratio:

$\boxed{ \bold{\: p_{A} : p_{B} = 3 : 1}}$