105-2019201 Practice

Question 1

Delta-v (literally "change in velocity"), symbolised as ∆v and pronounced delta-vee, as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such as launch from, or landing on a planet or moon, or in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of the vehicle.

When rocket thrust is applied in short bursts the other sources of acceleration may be negligible, and ∆v becomes

$\Delta V=\int _{t_{0}}^{t_{1}}\left|{\vec {a}}\right|dt$

As a simple example, take a conventional rocket which achieves thrust by burning fuel. Delta-v is the change in velocity that can be achieved by burning that rocket's entire fuel load.

(a)

Is it possible to calculate delta-v requirements of an orbital maneuver from conservation of energy by considering only the total energy of the vehicle in the initial and final orbits?

(b)

Orbit maneuvers are made by firing a thruster to produce a reaction force acting on the spacecraft. The size of this force will be

$F_{T}=v_{\text{exh}}\rho$

where $v_{\text{exh}}$ is the velocity of the exhaust gas in rocket frame and $\rho ={\frac {dm}{dt}}$ is the propellant flow rate to the combustion chamber.

Assuming $v_{\text{exh}}$ to be a constant not depending on the amount of fuel left prove the ∆v is given by Tsiolkovsky rocket equation.

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

where $m_{0}$ is the initial mass of the spacecraft (before the thrust), and $m_{1}$ is the final mass of the spacecraft (after the thrust).

solutions