Distance between the centres of two stars is `10alpha.` The masses of these stars are M and 16M and their radii a and 2a, respectively. A body of mass m is fired straight form the surface of the larger star towards the smaller star. What should be its minimum inital speed to reach the surface of the smaller star? Obtain the expression in terms of G,M and a.

Explanation

Let the gravitational field be zero at distance `x` from centre of star `A`. At `P`: Gravitational field due to `A`,
`E_(A)=(GM)/(x^(2))` towards left.
Gravitational field due to `B`,
`E_(B)=(G.16M)/((10a-x)^(2))` towards right.
Since net gravitational field at `P` is zero i.e.,
`E_(A)=E_(B)`
`(GM)/(x^(2))=(16M)/((10a-x)^(2))`
Taking square root
`1/x=4/((10a-x)) implies x=2a`
The body should be given velocity in such a manner so that on reaching at `P`, its velocity should be greater than zero. Total energy of body at the surface of `B`
`=1/2mv^(2)-(G(16M)m)/(2a)-(GM)/(8a)=1/2mv^(2)-(65GMm)/(8a)`
Total energy of body at `P`
`=1/2mv_(P)^(2)-(G(16M)m)/((10a-x))-(GMm)/x`
`1/2mv_(P)^(2)-(16GMm)/(8a)-(GMm)/(2a)`
`=1/2mv_(P)^(2)-(5GMm)/(8a)`
By conservation of energy
`1/2mv_(P)^(2)-(5GMm)/(2a)-1/2mv^(2)-(65GMm)/(8a)`
`1/2mv_(P)^(2)=1/2mv^(2)-(45GMm)/(8a)`
`v_(P)^(2)=v^(2)-(45GMm)/(4a)`
`v_(p)ge0`
`v^(2)-(45GMm)/(4a)ge0`
`v ge sqrt((45GMm)/(4a))`
`v_("min")=sqrt((45GMm)/(4a))`