B4. A particle of mass m is dropped vertically from rest from a height h > 0 above the surface of the Earth at z = 0. Gravity acts downwards in the negative 2 direction, giving a force of magnitude GMm (Z + a)? where a is the radius of the Earth, G is the gravitational constant and M is the mass of the Earth. F(z)= (a) Taking only the gravitational force into account (i.e. neglecting air resistance), write down the equation of motion and show that it can be written as a separable differential equation for the velocity v seen as a function of z. (b) By integrating the differential equation for 1), show that v2=2GM( l 1) z+a h+a Deduce the velocity of of the particle as it reaches the surface of the Earth z = 0. (c) Find a potential function U(z) for the gravitational force F(z). Hence, recover the previous result for vf using conservation of energy. (d) As it reaches the ground level, the particle enters the soil and is brought to rest by the action of a constant frictional force F opposing its movement in the z direction. Find the depth it goes into the ground in terms of F and of