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Two projectiles of the same mass have their maximum kinetic energies in ratio 4:1 and the ratio of their maximum heights is also 4:1 . Then what is the ratio of their ranges?
(A) 2:1
(B) 4:1
(C) 8:1
(D) 16:1

Answer
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Hint We are given here with the kinetic energy ratio and the ratio of maximum height and we are asked to find out the ratio of their ranges. So we will find the ratio of their velocities and angles of the projectile and use the formula for range.
Formula used
Ek=12mu2
Where, Ek is the kinetic energy of the projectile, m is the mass of the projectile and u is the initial velocity of the projectile.
H=u2sin2θ2g
Where, H is the maximum height of the projectile, u is the initial velocity of the projectile, θ is the angle of the projectile with the horizontal and g is the acceleration due to gravity.
R=u2sin2θg
Where, R is the range of the projectile, u is the initial velocity of the projectile, θ is the angle of the projectile with the horizontal and g is the acceleration due to gravity.

Complete Step By Step Solution
We are given,
Kinetic Energy of the first projectileKinetic Energy of the second projectile=Ek1Ek2=41
Thus, putting in the formula for kinetic energy, we can say
 12mu1212mu22=41
Thus, after cancellation, we get
u12u22=41
Thus, we get
u1u2=21u1:u2=2:1
Now,
Maximum Height Of the first projectileMaximum Height Of the second projectile=u12sin2θ12gu22sin2θ22g=41
After cancellation and Putting in u12u22=41 , we get
sin2θ1sin2θ2=11
Thus, we can say
θ1=θ2
Now,
Range of the first projectileRange of the second projectile=u12sin2θ1gu22sin2θ2g
After cancellation and putting in u12u22=41 and θ1θ2=11, we getRange of the first projectile: Range of the second projectile=4:1

Hence, the correct option is (B).

Note We evaluated the value of u1u2 and u12u22. This was for being more precise with the answer. Moreover, the value of u1u2 could±21. But the value of velocity of a projectile cannot be negative. Thus, we took the value of u1u2 to be 21.
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