Amass m is split into two parts, m and m-m, which are then separated by a certain distance. what ratio m/m maximizes the magnitude of the gravitational force between the parts

Ratio to maximize force = 1/1 The equation for gravitational attraction is F = G*m1*m2/r^2 where G = gravitational constant m1,m2 = mass of each mass r = distance between mass centers. Since the values of G and r are remaining constant as well as the sum of the masses. Let's simplify the problem as follows. m = mass 1 (1-m) = mass 2 So we're looking for a value of m that maximizes m(1-m) where m is somewhere between 0 and 1. Let's do it now: F = m(1-m) F = m - m^2 Now when you're looking to maximize a quantity, that screams "First derivative". So let's calculate that now. F = m - m^2 F' = 1 - 2m And set it to 0. F' = 1 - 2m 0 = 1 - 2m 2m = 1 m = 1/2 So the optimal value of m is 1/2. So m1 = 1/2 and m2 = 1 - 1/2 = 1/2. So the optimal ratio between the two masses to maximize the gravitational force is 1/1.