Two point masses M and 16M are fixed at a separation distance of 10a . A small particle is released from rest at the point on the line joining them where the net gravitational field is zero. What is the minimum speed the particle must be given at this point so that it escapes to infinity?

Step-by-step solution

Let the neutral point be at a distance x from the mass M . Equating the magnitudes of the gravitational fields due to both masses: GM x^2 = G(16M) (10a-x)^2 Taking the square root on both sides: 1 x = 4 10a-x 10a - x = 4x 5x = 10a x = 2a The distance of this neutral point from the mass 16M is 10a - 2a = 8a . The net gravitational potential V at this neutral point is: V = - GM 2a - G(16M) 8a V = - GM 2a - 2GM a = - 5GM 2a To escape to infinity, the total mechanical energy of the particle must be at least zero. By co