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Question:

Two spherical planets P and Q have the same uniform density ρ, masses Mp and Mq, and surface areas A and 4A, respectively. A spherical planet R also has uniform density ρ and its mass is (Mp + Mq). The escape velocities from the planets P, Q, and R are Vp, Vq, and Vr, respectively. Then

VRVP=3

VpVQ=12

VR>VQ>VP

None of these

Solution:

let mass of planet P=M and radius of planet P=R
Ap=A
Aq=4A
∴Rp=R
And Rq=R/2
Also, density=mass/volume
mass of planet Q=8M and radius of planet Q=2R
mass of planet R=9M and radius of planet R=9^(1/3)R
so escape velocity of P=Vp=√(2GMR)
Vq=√(2G8M/2R)=2Vp
Vr=√(2G9M/9^(1/3)R) = 9^(1/3)Vp
so Vr>Vq>Vp
and Vq/Vp=2