Physics

Problem

A spacecraft of mass

m

is launched radially outward from a planet of mass

M

and radius

R

. It leaves the planet’s surface with speed

u

and moves under gravity alone. (i) Obtain an expression for the change in gravitational potential energy of the spacecraft when it travels from

r_{1}=2R

r_{2}=4R

measured from the planet’s centre. (ii) Assuming no other forces act, determine the minimum initial speed

u_{\min}

r_{1}=2R

required for the spacecraft to reach

r_{2}=4R

with zero speed. (iii) If the actual launch speed is

1.5\,u_{\min}

, calculate the speed of the spacecraft when it reaches

r_{2}=4R