Question

Two bodies with kinetic energies in the ratio 2:3 are moving with equal momentum. Find the ratio of their masses?

Answer 1

Kinetic energy ( K.E ) of a body = 1/2 × m × v(Squared)
where m = mass of a body , v = velocity of a body
=> K.E = 1/2 × m × v × v = 1/2 × (mv) × v
On dividing and multiplying by mass m we get ,
=> K.E = 1/2 × (mv) × v × ( m / m )
=> K.E = 1/2 × (mv) × (mv) / m
=> K.E = 1/2 × (mv)^2 × 1/m
=> K.E = 1/2 × p^2 x 1/m -- ( i )
where p = mv ( momentum of a body )

Let k1 and k2 be kinetic energies of two bodies then we must have
k1 = 1/2 × (p1)^2 × 1/ m1 and k2 = 1/2 × (p2)^2 × 1 / m2
using equation ( i )
where , p1 = momentum of one body , p2 = momentum of other body , m1 = mass of one body and m2 = mass of other body.
Now given that , p1 = p2 = p , say ( since bodies have equal momentum ) and k1 : k2 = 2 : 3

k1 = 1/2 × (p)^2 × 1/ m1 and k2 = 1/2 × (p)^2 × 1/ m1
( since p1 = p2 = p )

Now , k1 : k2 = 2 : 3 => k1 / k2 = 2 /3
=> k1 = 2 /3 × k2
=> 1/2 × (p)^2 × 1/m1 = 2/3 × 1/2 × (p)^2 × 1/m2

Terms like 1/2 and (p)^2 cancels out from both sides

=> 1 / m1 = 2/3 × 1 / m2
Taking reciprocal on both sides we get ,
m1 = 3 /2 × m2
=> m1 / m2 = 3/2
=> m1 : m2 = 3 : 2