Question

two stones having masses in ratio 3:2 are dropped from heights in ratio 4:9 .the ratio of magnitudes of their linear momenta just before reaching the ground is ? (neglect air resistance)

Answer 1

m₁ = mass of first stone

m₂ = mass of second stone

Given that :   m₁ /m₂ = 3/2

h₁ = height from which mass "m₁" is dropped

h₂ = height from which mass "m₂" is dropped

given that : h₁ /h₂ = 4/9

v₁ = speed of mass "m₁" just before reaching the ground

v₂ = speed of mass "m₂" just before reaching the ground

velocity of an object from from height "h" is given as

v = sqrt(2gh)

hence

v₁ = sqrt(2gh₁)

and

v₂ = sqrt(2gh₂)

hence the ratio of velocities is given as

v₁ /v₂ = sqrt(2gh₁ /(2gh₂))

v₁ /v₂ = sqrt(h₁ /h₂)

v₁ /v₂ = sqrt(4/9)

v₁ /v₂ = 2/3

momentum is given as

P = m v

hence

P₁ = m₁ v₁

and

P₂ = m₂ v₂

taking the ratio of linear momenta

P₁ /P₂ = m₁ v₁/(m₂ v₂)

P₁ /P₂ = (m₁/m₂) (v₁/v₂)

P₁ /P₂ = (3/2) (2/3)

P₁ /P₂ = 1/1

hence ratio is 1:1