Solve Propulsion Problem: Rocket Mass, Fuel, Velocity, Altitude

[Dr.Brain]

I was doing some problems from my problem book and got stuck at this one:

A rocket of mass M has "fuel and oxidizers" inside it worth 'm' kgs . When propelling the exhaust gases have a constant speed of 'v' and the GASES are emitted at a constant rate of N kg/sec .Neglecting the air resistance effects.

(a)Calculate the equation for the trajectory of the rocket taking in consideration the effect of changing "g" with altitude "y".

(b) Calculate the altitude at which rocket will burn out?

(c) Height at which no external thrust due to "g" takes place.


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For the first part, I first derived an equation for variable mass system.

F(external thrust)=M dv/dt - dM/dt (V)

where V=relative velocity of rocket and the exhaust gases
M=initial mass of rocket along with fuels
dv/dt= gradual increase in velocity of rocket as it gains altitude

I guess here external thrust will be provided by "g" since it is the only external force acting on the rocket-fuel system.

So what I did was:

F(external)=M(t) g(t) (because both mass of the rocket and value of g will be changing with altitude)

Therefore, F(external) => M(t)g(t)=M dv/dt-dM/dt ( dy/dt-v)

Beacuse "v" is the constant velocity of exhaust gases and dy/dt represent the changing velocity of the rocket .

Now I write dv/dt = differential of dy/dt

and also g(t)= g [R/y+R]^2 ( from Gravitation chapter)

And now i get a complex equation in which double integration is required .I am stuck here.Please help me if anyone here can get me a simpler metod.

 

[OlderDan]

I'm not sure this is the right approach, but it occurs to me that perhaps you should formulate the problem in terms of work and energy. The expelled fuel provides a thrust to the rocket that is time independent I believe. The work done by the fuel will increase the rocket energy. Part c) suggests to me that they are talking about achieving orbit. g never goes to zero, so they must be talking about when g is just sufficeint to provide the centripetal acceleration required for an orbit. And that suggests that you need to be taking into consideration the initial angular momentum and kinetic energy of the rocket at launch time.

Many trajectories are possible, so there must be some assumption about the direction of motion and direction of thrust. My first guess would be to require vertical thrust, which does not mean the rocket goes straight up relative to a point on the earth. If you want the rocket to always be above a point on the earth, then your thrust is going to have a horizontal component. This problem smacks of "real world" rocketry. I'm not sure what the assumptions should be about the trajectory.

 

[Dr.Brain]

I am already formulating this problem in terms of work and energy.I have actually derived the variable mass eqn from the fact that change in momentum is due to an impulse due to external forces and the only possible external force possible is that of gravity force which will always be vertical till a time comes when the rocket escapes Earth gravity and moves freely.I think that will be the point at with F(external)=0 as in part (c). I think either i have made this problem a bit complicated or maybe i am stuck at the end integrals that i am not able to solve.