Question

If momentum of two bodies are equal, will their kinetic energy [K.E.] be also equal?

Explain with complete calculations & justifications.

Answer 1

let two bodies of masses m and M move with the velocity u and U .

we know ,
momentum of any object =mass of object x velocity of that object

so, momentum of 1st body =mu
momentum of 2nd body =MU
a/c to question ,
mu =MU
m/M = U/u -------------(1)

now,
kinetic energy of 1st body =1/2 mu^2
kinetic energy of 2nd body =1/2MU^2

K.E of 1st body/kinetic of 2nd body
=1/2mu^2/1/2MU^2
=(m/M)(u/U)^2 =(m/M)(M/m)^2
= (M/m)

e.g.

m .K.E of first body =M KE of 2nd body .

so, statement if momentum of two bodies is equal then their kinetic energy will be also equal is false .

because kinetic energy =p^2/2m
means kinetic energy not only depend on P also m .
if P is constant then kinetic energy is inversely proportional to mass of body .

Answer 2

p = m v
v = p/ m

now
KE = 1/2 m v^2
KE = 1/2 m (p/m)^2
KE = 1/2 x m x p^2/ m^2
KE = p^2/ 2m

let the mass of the two bodies be
m1 and m2 respectively

KE of the body of mass m1
= p^2 / m1

KE of the body of mass m2
= p^2 / m2

here we see that the mass of both the bodies are different
therefore the KE of the bodies are also not equal