Question

Two masses 10gm and 40gm are moving with kinetic energies in the ratio 9:25.
The ratio of their linear momenta is
1) 5:6
2) 3:10
3) 6:5
4) 10:3​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{P_{1}:P_{2}=3:10}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given :}} \\ \tt: \implies Two \: mases = 10 \: gm \: and \: 40 \: gm \\ \\ \tt: \implies Ratio \: of \: kinetic \: energy = 9 :25 \\ \\ \red{\underline \bold{To \: Find :}} \\ \tt: \implies Ratio \: of \: momentum =?$

• According to given question :

$\bold{As \: we \: know \: that} \\ \tt: \implies \frac{K.E_{1} }{ K.E_{2} } = \frac{9}{25} \\ \\ \tt: \implies \frac{ \frac{1}{2} m_{1} (v_{1})^{2}}{\frac{1}{2} m_{2} (v_{2})^{2}} = \frac{9}{25} \\ \\ \tt: \implies \frac{2 m_{1}( { v_{1} })^{2} }{2 m_{2} ({ v_{2} })^{2}} = \frac{9}{25} \\ \\ \tt: \implies \frac{10 \times{ (v_{1} })^{2} }{40 \times { (v_{2} })^{2}} = \frac{9}{25} \\ \\ \tt: \implies \frac{({ v_{1} })^{2} }{4({ v_{2} })^{2}} = \frac{9}{25} \\ \\ \tt: \implies \frac{{ v_{1} }^{2} }{{ v_{2} }^{2}} = \frac{9 \times 4}{25} \\ \\ \tt: \implies (\frac{v_{1} }{v_{2}})^{2} = \frac{36}{25} \\ \\ \tt: \implies \frac{{ v_{1} }}{{ v_{2} }} = \sqrt{ \frac{36}{25} } \\ \\ \green{\tt: \implies \frac{{ v_{1} }}{{ v_{2} }} = \frac{6}{5} }$ $\\\bold{As \: we \: know \: that} \\ \tt: \implies \frac{ P_{1}}{ P_{2} } = \frac{ m_{1} v_{1} }{m_{1} v_{2}} \\ \\ \tt: \implies \frac{ P_{1}}{ P_{2} } = \frac{10 \times 6}{40 \times 5} \\ \\ \green{\tt: \implies \frac{ P_{1}}{ P_{2} } = \frac{3}{10} } \\\\\green{\tt \therefore Ratio \: of \: momentum \: is \:3: 10}$

Answer 2

GIVEN :–

• Two masses 10gm and 40gm are moving with kinetic energies in the ratio 9:25.

TO FIND :–

Ratio of liner moment = ?

SOLUTION :–

• We know that kinetic energy is –

$\\ \longrightarrow \: { \boxed{ \bold{K.E. = \frac{1}{2}m {v}^{2} }}}$

• According to the question –

$\\ \implies\: { \bold{ \dfrac{(K.E.) _{1}}{(K.E.) _{2}} = \dfrac{9}{25} }}$

$\\ \implies\: { \bold{ \dfrac{ \dfrac{1}{2}m_{1}v {}^{2} _{1}}{\dfrac{1}{2}m_{2}v {}^{2} _{2}} = \dfrac{9}{25} }}$

$\\ \implies\: { \bold{ \dfrac{ (10)v {}^{2} _{1}}{(40)v {}^{2} _{2}} = \dfrac{9}{25} }}$

$\\ \implies\: { \bold{ \dfrac{ v {}^{2} _{1}}{4v {}^{2} _{2}} = \dfrac{9}{25} }}$

$\\ \implies\: { \bold{ \dfrac{ v {}^{2} _{1}}{v {}^{2} _{2}} = \dfrac{9 \times 4}{25} }}$

$\\ \implies\: { \bold{ \dfrac{ v _{1}}{v _{2}} = \sqrt{\dfrac{36}{25}} }}$

$\\ \implies\: \large { \boxed{ \bold{ \dfrac{ v _{1}}{v _{2}} = {\dfrac{6}{5}} }}}$

• We know that Liner momentum –

$\\ \longrightarrow \: \large { \boxed{ \bold{P = mv }}}$

• So that , ratio of Liner moment –

$\\ \implies \: { \bold{ \dfrac{P _{1}}{P _{2}} = \dfrac{m_{1} v_{1}}{m_{2} v_{2}} }}$

$\\ \implies \: { \bold{ \dfrac{P _{1}}{P _{2}} = \left( \dfrac{m_{1}}{m_{2} } \right) \left( \dfrac{v_{1}}{v_{2} } \right)}}$

• Now put the values –

$\\ \implies \: { \bold{ \dfrac{P _{1}}{P _{2}} = \left( \dfrac{10}{4 0 } \right) \left( \dfrac{6}{5} \right)}}$

$\\ \implies \: { \bold{ \dfrac{P _{1}}{P _{2}} = \dfrac{60}{200 } }}$

$\\ \implies \: \large { \boxed{ \bold{ \dfrac{P _{1}}{P _{2}} = \dfrac{3}{10 } } }}$

▪︎ Hence , Option (B) is correct.