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

Distance between the centres of two stars is 10a. The masses of these

1.	Distance between the centres of two stars is 10a. The masses of these star are M and 16 M and their radii a and 2a, respectively. A body of mass m is fired straight from the surface of the largr star towards the smaller star. What should be its minimum initial speed to reach the surface of the smaller star ? Obtain the expression in terms of G, M and a.
Answer» Solution :The distance x (from the smaller planet) where the gravitational pull of two planet.s balance each other will be given by `(-GMm)/(x^(2)) = (-G(16M))/((10a-x)^(2))` i.e., So the body will REACH the smaller planet due to planet.s gravitational field if it has SUFFICIENT energy to CROSS the point B(x=2a), i.e., `(1)/(2) mv^(2) gt m(V_(B)-V_(S))` But `V_(S) = -[(16 GM)/(2a) + (GM)/((10-2a))] = (65 GM)/(8a)` and `V_(B) = -[(16 GM)/(8a) + (GM)/(2a)] = -(20 GM)/(8a)` so `(1)/(2) mv^(2) gt m [(65 GM)/(8a) - (20 GM)/(8a)]` i.e., `v_("min") = (3)/(2)sqrt((5GM)/(a))`