Unusual difficult kinetic problem

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SUMMARY

The discussion focuses on solving a complex kinetic problem involving a rocket's mass and momentum changes due to exhaust gases. Participants emphasize the use of conservation of momentum and energy balance, specifically through the integral form of the kinetic energy equation. The key equation presented is the integral of mass times velocity, ∫ m(t) v dv = qm, which leads to a differential equation essential for deriving the required formula. The approach involves analyzing the initial and final momentum of the rocket and exhaust gases to establish a relationship between them.

PREREQUISITES
  • Understanding of general kinetics formulas
  • Familiarity with conservation of momentum principles
  • Knowledge of differential equations
  • Basic concepts of rocket propulsion and mass flow
NEXT STEPS
  • Study the derivation of the rocket equation using conservation of momentum
  • Learn about energy balance in dynamic systems
  • Explore the application of differential equations in physics problems
  • Investigate the integral form of kinetic energy equations
USEFUL FOR

Students in physics or engineering, particularly those studying dynamics and rocket propulsion, as well as educators looking for problem-solving strategies in kinetic theory.

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Homework Statement


http://img80.imageshack.us/img80/1797/20799357.png

Homework Equations


General kinetics formulas

The Attempt at a Solution


Actually I don't know how to start. The initial mass is M when qm energy is used the mass is M-m. If I use an energy balance I must use the integral version, right?

\int m(t) v \mbox{d}v = qm

But I don't know how to start.
 
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I myself have never done this before, but I will try to help you. I think my way should get the required formula.

Consider the exhaust gases (a small amount 'dm') at time 't'.

Initial momentum pt= dm*ve

Final momentum pt+dt = dm(vR-ve)

Rocket:

pt=(M-dm)vR
pt+dt= (M-dm)(vR-ve).

Now use conservation of momentum here: Loss in change of momentum in gas = change momentum gained by rocket

Using this, you should now be able to get a differential equation to solve (remember, 'dm' is infinitesimally small)
 

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