Distance between the centre of two stars is 10a. the masses of these stars are M and 16M and their radii a and 2a respectively. A body of mass m is fired straight from the surface of the larger star towards the surface of the smaller star. What should be its minimum initial speed to reach the surface of the smaller star ?

Let there are two stars 1 and 2 as shown below.Let P is a point between C1 and C2 , where gravitational field strength is zero. Hence GMr12=G(16M)r22;r2r1=4, r1+r2=10a∴ r2=44+1(10a)=8ar1=2aNow, the body of mass m is projected from the surface of large star towards the smaller one. Between C2 and P it is attracted towards 2 and between C1 and P it will be attracted towards 1.Therefore, the body should be projected to just cross point P because beyond that the particle is attracted towards the smaller star itself.From conservation of mechanical energy 12mv2= potential energy of the body at P-potential energy at the surface of larger star.∴12mvmin2=GMmr1−16GMmr2−−GMm10a−2a−16GMm2a12mvmin2=458GMmavmin=352GMa