Re: Rocket science!

*To*: mathgroup at smc.vnet.net*Subject*: [mg23864] Re: Rocket science!*From*: Hendrik van Hees <h.vanhees at gsi.de>*Date*: Thu, 15 Jun 2000 00:51:04 -0400 (EDT)*Organization*: GSI Darmstadt*References*: <8i1t1n$jmc@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Anders Andersen wrote: > > Hi! > > I'm trying to solve a partial differential equation for the velocity and > mass of a rocket, but i don't know how do this in Mathematica (3.0). The > equation is: > > m*(dv/dt) = -m*g - u*(dm/dt) > > Where d is the derivative, g = 9.8 m/s and u is som velocity less than about > 5000 m/s. The start velocity is v(0) = 0 and the start mass is m(0) = > "initial mass of rocket including fuel". You've to write v*(dm/dt) on the rhs of your equation. It is of course noting than Newton's 1st law of mechanics in its only correct and complete form, namely dp/dt=-m*g As your example nicely shows the usually used description m dv/dt on the right hand side is only true if m is const. In Mathematica you only need to call DSolve[m[t] D[v[t],t]==-m[t] g-v[t] D[m[t],t],v[t],t] This tells Mahematica to solve the differential equation in the first entry to the function v[t] while the independent variable is t. The result is a list with substitution rules for (hopefully all) the solutions of your differential equation. C[1] g Integrate[m[DSolve`t], {DSolve`t, 0, t}] {{v[t] -> ---- - ------------------------------------------}} m[t] m[t] In your case it's a linear differential equation of first order and according to general theorems about these type of ode's it has only this solution. C[1] is a integration constant which can be fixed with a given initial condition. Suppose you use the initial condition v[0]=0 then you can call DSolve as follows: DSolve[{m[t] D[v[t],t]==-m[t] g-v[t] D[m[t],t],v[0]==0},v[t],t] which leads to the correct solution g Integrate[m[DSolve`t], {DSolve`t, 0, t}] {{v[t] -> -(------------------------------------------)}} m[t] -- Hendrik van Hees Phone: ++49 6159 71-2751 c/o GSI-Darmstadt SB3 3.183 Fax: ++49 6159 71-2990 Planckstr. 1 mailto:h.vanhees at gsi.de D-64291 Darmstadt http://theory.gsi.de/~vanhees/index.html