Answer:
(a) h = 1.27 Re
(b) h = 1.86 Re
Explanation:
Let M is the mass of earth and Re is the radius of earth.
initial velocity of projection, v = 0.462 ve
where, ve is the escape velocity of an object on earth surface.
(a)
The value of escape velocity is

So,
.... (1)
By using conservation of energy
(Kinetic energy + potential energy ) at the surface of earth = Potential energy at the height h.
where, h is the maximum height upto which the projectile reach
K.E at surface P.E at surface = P.E at the top

By equation (1), substituting the value of v


h = 1.27 Re
(b)
initial kinetic energy = 0.462 times the kinetic energy required to escape


So, again by using the conservation of energy
Kinetic energy at the surface + Potential energy at the surface = Potential energy at the top


h = 1.86 Re