## Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (3rd Edition)

The energy required to boost the shuttle to the new orbit is $1.16\times 10^{11}~J$.
We can find the change in mechanical energy when the shuttle moves to a higher orbit. Let $M_e$ be the earth's mass and let $M_s$ be the shuttle's mass. Let $R$ be the earth's radius. $\Delta E_{mech} = \frac{1}{2}\Delta U$ $\Delta E_{mech} = \frac{1}{2}(U_f-U_0)$ $\Delta E_{mech} = \frac{1}{2}[-\frac{G~M_e~M_s}{R+610~km}-(-\frac{G~M_e~M_s}{R+250~km})]$ $\Delta E_{mech} = \frac{1}{2}(\frac{G~M_e~M_s}{R+250~km}-\frac{G~M_e~M_s}{R+610~km})$ $\Delta E_{mech} = \frac{1}{2}(G~M_e~M_s)(\frac{1}{R+250~km}-\frac{1}{R+610~km})$ $\Delta E_{mech} = \frac{1}{2}(6.67\times 10^{-11}~m^3/kg~s^2)(5.98\times 10^{24}~kg)(75,000~kg)(\frac{1}{6.63\times 10^6~m}-\frac{1}{6.99\times 10^6~m})$ $\Delta E_{mech} = 1.16\times 10^{11}~J$ The energy required to boost the shuttle to the new orbit is $1.16\times 10^{11}~J$.