Derive the Breguet range formula which is;

Range $=S=I_{sp}{V}_{}(\frac{L}{D})ln(\frac{M_{\text{t}\text{a}\text{k}\text{e}\text{o}\text{f}\text{f}}}{M_{\text{F}\text{i}\text{n}\text{a}\text{l}}})$

Assume that we cruise, with no acceleration, with a constant $V_{},(\frac{L}{D})$ and $I_{sp}$

$=$ thrust $D=$ drag $L=$ lift,,$W=$ weight

$M_{\text{t}\text{a}\text{k}\text{e}\text{o}\text{f}\text{f}}=$ takeoff mass

$M_{\text{F}\text{i}\text{n}\text{a}\text{l}}=$ final vehicle mass after fuel is consumed

a$)$ Set $V_{}=$ flight velocity $=d\frac{x}{d}t$

b$)$ Set $m_{\text{f}\text{u}\text{e}\text{l}}^{}=$ fuel mass flow rate $=-\frac{dM}{dt},$ this is conservation of mass

$I_{sp}=$ specific impulse defined to be: $=\frac{J}{m_{\text{f}\text{u}\text{e}\text{l}}^{}*(g)},g=9\mathrm{.}8\frac{m}{s^2}$

Hint: start with $J=D.$ Replace thrust with $I_{sp},$ remove dt using the two formulas that

contain dt