Question

Two bodies of mass m and 4m have linear momenta in the ratio 2:1. What is the ratio of their kinetic energy

Answer 1

Linear momentum and kinetic energy can be related as:-
P=[2m(K.E.)]^1/2 where 'm' is the mass and K.E. is the kinetic energy of the body.
Let say P1 & P2 be linear momentum of mass 'm' & '4m' resp.
P1 = [2m(2K.E.)]^1/2
P2= [2(4m)(K.E.)]^1/2
P1/P2 = (1/2)^1/2


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Answer 2

mass of 1st body, m₁ = m

velocity of 1st body, v₁ = v₁

mass of 2nd body, m₂ = 4m

velocity of 2nd body, v₂ = v₂

∴ratio of linear momenta of the 2 bodies, p₁/p₂ = m₁v₁/m₂v₂

= mv₁/4mv₂ = v₁/4v₂

⇒ 2/1 = v₁/4v₂

⇒  8v₂ = v₁

∴ Ratio of kinetic energies = (1/2m₁v₁²) / (1/2m₂v₂²) = m×64×v₂²/4m×v₂² = 64/4 = 16:1