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## Homework Statement

Consider a rocket traveling in a straight line subject to an external force Fext acting along the

same line.

a) Show that the equation of motion is [itex]m\dot{v}=-\dot{m}v_{ex} + F^{ext}[/itex] (1)

b) Specialize to the case of a rocket taking off vertically (from rest) in a (constant) gravitational

field g, so the equation of motion becomes

[itex]m \dot{v} = -\dot{m} v_{ex} -mg[/itex] (2)

Assume that the rocket ejects mass at a constant rate, [itex]\dot{m}=-k[/itex] (where k is a positive constant), so that [itex]=m = m_0 - kt[/itex]. Solve the equation for v as a function of t.

(c) Using the rough data from problem 3.7, find the space shuttle's speed two minutes into flight, assuming (what is nearly true) that is travels vertically up during this period and that g does not change appreciably. Compare with the corresponding result if there were no gravity. (d) Describe what would happen to a rocket that was designed so that the first term on the right of equation (2) was smaller than the initial value of the second.

## Homework Equations

## The Attempt at a Solution

I was successful in solving part (a). Now I am trying to integrate (2), of which I successfully derived when I derived equation (1) on my own. Here is my work:

[itex]m \dot{v} = -\dot{m} v_{ex} -mg[/itex] Since we are dealing with objects of mass, we can safely assume that [itex]m \ne 0[/itex], whereby we can divide by m:

[itex] \dot{v} = - \frac{\dot{m}}{m} v_{ex} -g[/itex].

Integrating with respect to time,

[itex]v(t) = -v_{ex} \int_?^?\frac{\dot{m}}{m} +v_0 - gt[/itex] At first I suspected the limits of integration to be m0 to m, where m0 is the initial mass and m is the mass at some time t. This didn't seem correct, however. Isn't this integration integrating over the amount of that has left the rocket? And if we were to integrating with the lower bound m0, would that imply that initially all of the mass of the rocket has gone into fuel? I am having difficulty interpreting the limits of integration. If someone could be so kind as to help me, I would certainly appreciate it.

Thanks y'all.

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