Question

two objects of mass 1kg and 4 kg have equal momentum what is the ratio of there kinetic energy ?​

Answer 1

Answer:

KE1:KE2 = v1:v2

Step-by-step explanation:

I guess this is the answer

Answer 2

$\huge ✯ { \underline{ \underline{\bold{ ᴀɴsᴡᴇʀ }}}} ✯$

★ given →

☞︎︎︎ two objects of mass 1 kg and 4 kg have equal momentum

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★ to find →

☞︎︎︎ ratio of kinetic energy possessed by two objects

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★ solution →

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first we have to derive relation between kinetic energy and momentum!

• kinetic energy = K.E
• momentum = M
• mass = m
• velocity = v

kinetic energy : work done by an object due to its Motion is called kinetic energy

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

momentum : product of mass and velocity of an object is called momentum

$\boxed{ \pink{ \bold{M = mv}}}$

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we know that,

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

now, multiply m numerator and denominator of LHS

$\bold{ \implies K.E = \frac{m {v}^{2} \times m}{2 \times m} } \\ \\ \bold{ \implies K.E = \frac{ {m}^{2} {v}^{2} }{2m} } \: \: \: \: \: \: \: \: \\ \\ \bold{ \implies K.E = \frac{ {(mv)}^{2} }{2m} } \: \: \: \: \: \: \:$

and we know that,

mv = M, momentum

$\bold{ \implies \:K.E = \frac{ {M}^{2} }{2m} } \: ......(i)$

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now, we have

• mass of first object m1 = 1kg
• mass of second object m2 = 4kg
• momentum of both object is equal

➪ M1 = M2 .....(ii)

☆ kinetic energy possessed by object first, K.E1

from equation (i)

${ \tt{ \implies \:K.E = \frac{ {M}^{2} }{2m} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \\ { \tt \implies \: K.E 1 = \frac{ M1^{2} }{2m1} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \tt \implies \: K.E 1 = \frac{ M1^{2} }{2 \times 1}} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \tt \implies \boxed{ { \tt\red{ K.E 1 = \frac{ M1^{2} }{2}}}} }\: \: .........(iii)$

☆ kinetic energy possessed by object second, K.E2

from equation (i)

${ \tt{ \implies \:K.E = \frac{ {M}^{2} }{2m} }} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ { \tt{ \implies \:K.E2= \frac{ {M2}^{2} }{2m2} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \\ { \tt{ \implies \:K.E2 = \frac{ {M2}^{2} }{2 \times 4} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ \\{ \tt{ \implies \boxed{ { \tt\red{ \:K.E2 = \frac{ {M2}^{2} }{8}}}}}} \: ..........(iv)$

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now, we have to find ratio of kinetic energy possessed by both objects.

➪ K.E1 : K.E2

${\tt ratio \: of \: K.E1 \: and \: K.E2 \: = \frac{K.E1}{ K.E2} }$

now, put tha value of K.E1 and K.E2 from eq. (iii) and (iv)

${ \tt \implies \: \frac{K.E1}{K.E2} = \frac{ {M1}^{2} }{2} \div \frac{ {M2}^{2} }{8}} \\ \\ { \tt \implies \frac{K.E1}{K.E2} = \frac{{M1}^{2}}{2} \times \frac{8}{{M2}^{2}} }$

from equation (ii) we know that,

K.E1 = K.E2

${ \tt \implies \frac{K.E1}{K.E2} = \frac{\cancel{{M1}^{2}}}{2} \times \frac{8}{ \cancel{{M1}^{2}} }} \: \: \: \: \: \: \: \: \\ \\ { \tt \implies \frac{K.E1}{K.E2} = \cancel{ \frac{8}{2} } = \frac{4}{1} } \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \\ \\ \implies \boxed{ \orange{\bold{ K.E1 \: : \: K.E2 = 4 :1 }} }$

hence, ratio of their kinetic energy is 4:1