Energy Conservation

• Approach #1

  • Start at $v=0$
  • $$\begin{array}{rl}\Delta KE& =\frac{1}{2}(M-\Delta m)(-\Delta v{)}^2+\frac{1}{2}\Delta m{u}_e^2-0\\ \Delta KE& =\frac{1}{2}M\Delta v^2-\frac{1}{2}\Delta m\Delta v^2+\frac{1}{2}\Delta m{u}_e^2\\ \to \frac{\Delta KE}{\Delta t}& =\frac{1}{2}M(\frac{\Delta v}{\Delta t})\Delta v+\frac{1}{2}\frac{\Delta m}{\Delta t}{u}_e^2\\ \underset{\Delta t\to 0}{\lim }⟹P& =\frac{1}{2}\dot{m}{u}_e^2\end{array}$$

• Approach #2

  • Start at $v\ne 0$
  • $$\begin{array}{rl}\Delta KE& =\frac{1}{2}(M-\Delta m)(v-\Delta v{)}^2+\frac{1}{2}\Delta m{u}_e^2-\frac{1}{2}M{v}^2\\ \to \frac{\Delta KE}{\Delta t}& =Mv(\frac{\Delta v}{\Delta t})+\frac{1}{2}\frac{\Delta m}{\Delta t}{u}_e^2-\frac{\Delta m}{\Delta t}{u}_{e}v\\ \underset{\Delta t\to 0}{\lim }⟹P& =Mv\frac{dv}{dt}+\frac{1}{2}\dot{m}{u}_e(u_e-2v)\end{array}$$
  • Power to Vehicle: $Mv\frac{dv}{dt}$
  • Power to Propellant: $\frac{1}{2}\dot{m}{u}_e(u_e-2v)$

IMPORTANT

Jet Power Equation

• $P_{jet}=\frac{1}{2}\dot{m}{u}_e^2$
• Arises from both approaches
• Rate at which energy goes into creating the exhaust stream
• Minimum power that must be supplied