Question

2 bodies with k.E in ratio 4:1 are moving with equal linear momentum .the ratio of their masses is

Answer 1

Refer to my above picture

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Answer 2

let the masses be m1 and m2, velocities be v1 and v2, kinetic energies be k1 and k2 and momentums be p1 and p2

so,
k1/k2 = 4/1
1/2m1(v1²) / 1/2m2(v2²) = 4/1
m1(v1)²/m2(v2²) = 4/1
m1/m2 = 4(v2²)/(v1²) - (i)

also
p1 = p2
m1v1 = m2v2
m1/m2 = v2/v1 - (ii)

from (i) and (ii)
4(v2²)/(v1²) = v2/v1
4(v2)/v1 = 1
4(v2) = v1

putting v1 = 4(v2) in (ii)
m1/m2 = v2/v1
m1/m2 = v2/4(v2)
m1/m2 = 1/4

so, ratio of masses is 1:4.