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Old Jan 8th 2018, 03:30 PM   #1
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Sticky rocket problem

I'm trying to get my head around problem 3.10 from John R. Taylor's book on Classical Mechanics:

3.10 * Consider a rocket (initial mass $\displaystyle m_o$) accelerating from rest in free space. At first, as it speeds up,
its momentum p increases, but as its mass m decreases p eventually begins to decrease. For what value of m is p maximum?
Earlier in the book he gives these three equations and which appear to be the only pertinent ones:
  • (3.4) $\displaystyle dP = m \: dv + dm \: v_{ex} = F_{ext} dt$
  • (3.5, 3.7) $\displaystyle m \dot{v} = -\dot{m}v_{ex}$
  • (3.8) $\displaystyle v-v_0 = -v_{ex} \: ln(\frac{m}{m_0})$

(3.4, 3.5, 3.7, 3.8) are the reference numbers of those equations in the book.

where m0 is the initial mass of the rocket (including the mass of fuel) and Vex is the velocity of the propellant coming out of the rocket. dm is the mass of the propellant exiting the rocket and v is the forward velocity of the rocket with v0 being the initial velocity at time t = 0.

The question seems to be asking for momentum (p) purely in terms of mass (m) and constants. However my attempts always seem to end up with a velocity (v) term or time (t) term and according to the text book this is supposed to be an easy problem. Any suggestions?
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