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Question:

Two projectiles of same mass have their maximum kinetic energies in ratio 4:1 and ratio of their maximum heights is also 4:1. Then what is the ratio of their ranges?

2:1

8:1

4:1

16:1

Solution:

Given : m1 = m2 = m
Let the projectiles are projected with respective velocity u1 and u2.
K.E1/K.E2 = 4/1
(1/2)mu1^2 / (1/2)mu2^2 = 4/1
=> u1 = 2u2. (1)
Maximum height attained by the projectile H = u^2sin^2θ/2g
Also it is given that H1:H2 = 4:1
=> u1^2sin^2θ1 / u2^2sin^2θ2 = 4/1
4u2^2sin^2θ1 / u2^2sin^2θ2 = 4/1
=> θ1 = θ2 (using (1))
Range of the projectile R = u^2sin2θ/g
=> R1/R2 = u1^2sin2θ1 / u2^2sin2θ2
R1/R2 = 4u2^2sin2θ2 / u2^2sin2θ2 = 4/1