Gravitation Question 125
Question: The masses and radii of the earth and moon are $ [M_{1},,R_{1}] $ and $ [M_{2},,R_{2}] $ respectively. Their centres are distance d apart. The minimum velocity with which a particle of mass m should be projected from a point midway between their centres so that it escapes to infinity is [MP PET 1997]
Options:
A) $ [2\sqrt{\frac{G}{d}(M_{1}+M_{2})}] $
B) $ [2\sqrt{\frac{2G}{d}(M_{1}+M_{2})}] $
C) $ [2\sqrt{\frac{Gm}{d}(M_{1}+M_{2})}] $
D) $ [2\sqrt{\frac{Gm(M_{1}+M_{2})}{d(R_{1}+R_{2})}}] $
Show Answer
Answer:
Correct Answer: A
Solution:
Gravitational potential at mid point $ [V=\frac{-GM_{1}}{d/2}+\frac{-GM_{2}}{d/2}] $ Now, $ [PE=m\times V=\frac{-2Gm}{d}(M_{1}+M_{2})] $ [m = mass of particle] So, for projecting particle from mid point to infinity $ [KE,=,|,PE,|] $ $ [\Rightarrow ,\frac{1}{2}mv^{2}=\frac{2,Gm}{d}(M_{1}+M_{2})] $ $ [\Rightarrow ,v=2\sqrt{\frac{G,(M_{1}+M_{2})}{d}}] $
