What is the change in momentum between the initial and final points of the trajectory path, if the range is maximum?
If we know that the range is maximum. The angle of projection, ϴ=45o. Now, since the initial velocity horizontally remains the same throughout the horizontal part, there is no change in momentum in the horizontal.  The magnitude of vertical velocity = u sin 45o, where ‘u’ is the initial velocity of the object.  The vertical velocity will be equal in magnitude but opposite in direction at the required points. Hence, the change in momentum = 2mu * sin 45o = √2mu  , where ‘m’= mass of the object. So the change in momentum will be the product of the square of two and mass and initial velocity.