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Re: Rocket science!

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


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

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] -> -(------------------------------------------)}}
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
D-64291 Darmstadt

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