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

• A. $2 \sqrt { \frac { G } { d } \left( M _ { 1 } + M _ { 2 } \right) }$
• B. $2 \sqrt { \frac { 2 G } { d } \left( M _ { 1 } + M _ { 2 } \right) }$
• C. $2 \sqrt { \frac { G m } { d } \left( M _ { 1 } + M _ { 2 } \right) }$
• D. $2 \sqrt { \frac { G m \left( M _ { 1 } + M _ { 2 } \right) } { d \left( R _ { 1 } + R _ { 2 } \right) } }$

Answer 1

Answer: (A)

$2 \sqrt { \frac { G } { d } \left( M _ { 1 } + M _ { 2 } \right) }$

Answer 2

Gravitational potential at mid point

$V = \frac { - G M _ { 1 } } { d / 2 } + \frac { - G M _ { 2 } } { d / 2 }$

Now, $P E = m \times V = \frac { - 2 G m } { d } \left( M _ { 1 } + M _ { 2 } \right)$

[m = mass of particle]

So, for projecting particle from mid point to infinity

$K E = | P E |$

$\Rightarrow \frac { 1 } { 2 } m v ^ { 2 } = \frac { 2 G m } { d } \left( M _ { 1 } + M _ { 2 } \right)$ $\Rightarrow v = 2 \sqrt { \frac { G \left( M _ { 1 } + M _ { 2 } \right) } { d } }$