(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

With what speed should a rocket be fired upward so that it never returns to earth? (Neglect all forces except the earth's gravitational attraction)

2. Relevant equations

The ONLY thing gone over so far involving rockets is the following formula:

Let the altitude of the rocket at time t be r(t), the mass be m(t) and the velocity of the exhaust matter, relative to the rocket be c(t). Then

[itex]m(t)r''(t)=-m'(t)c(t)-m(t)g[/itex]

This is in a section focusing on differential equations.

3. The attempt at a solution

Solving for r'':

[itex]r''(t) = - \frac{m'(t)}{m(t)} c(t) - g[/itex]

And now I don't really know where to go. I can't operate directly on this without making some assumptions about m and c, and based on the problems that we have worked on so far the procedure was similar to this:

Assume c(t) is constant, so c(t)=-c.

Let w be the initial weight of the rocket and fuel. Let k be the rate at which fuel is consumed. Then

[itex]m(t)=\frac{w-kt}{g}[/itex] and [itex]m'(t) = -\frac{k}{g}[/itex].

Now we have

[itex]r''(t)=\frac{kc}{w-kt}-g[/itex]

Integrating and using the initial condition r'(0)=0:

[itex]r'(t)=-c\ln(\frac{w-kt}{w})-gt[/itex]

Integrating again and using the initial condition r(0)=0:

[itex]r(t)=\frac{c(w-kt)}{w}ln(\frac{w-kt}{w})-\frac{1}{2}gt^2+ct[/itex]

It seems to me that, for this question, this model will not work. No matter what, gravity will pull back down. I really think I need to use a formula for gravity dependent on r(t) in order for this to work. Incidentally, the answer in the book is 6.96 mi/sec.

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# Homework Help: Velocity of a rocket which never returns to earth

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