105-2019201 Solutions

Question 1

(a)

It is not possible to determine delta-v requirements from conservation of energy by considering only the total energy of the vehicle in the initial and final orbits since energy is carried away in the exhaust. For example, most spacecraft are launched in an orbit with inclination fairly near to the latitude at the launch site, to take advantage of the Earth's rotational surface speed. If it is necessary, for mission-based reasons, to put the spacecraft in an orbit of different inclination, a substantial delta-v is required, though the specific kinetic and potential energies in the final orbit and the initial orbit are equal.

(b)

$a={\frac {F}{m}}=v_{\text{exh}}{\frac {\rho }{m}}$

$\Delta v=\int _{t_{0}}^{t_{1}}v_{\text{exh}}{\frac {\rho }{m}}dt$ $=-\int _{m_{0}}^{m_{1}}v_{\text{exh}}{\frac {1}{m}}{\frac {dm}{dt}}dt$ $=-\int _{m_{0}}^{m_{1}}v_{\text{exh}}{\frac {1}{m}}dm$

$\Delta v=v_{\text{exh}}\ln \left({\frac {m_{0}}{m_{1}}}\right)$