Other Calculations with Rockets

Other Calculations with rockets:

From Halliday, Resnick & Walker, Fundamentals of Physics – Extended, 5th Edition.

The acceleration of the rocket can be determined from the law of Conservation of Momentum.

Pi = Pf

Where the momentum is equal to M (the total mass of the system) times v (velocity). Initially, the rocket has a momentum of Mv. At some time later, as the hot exhaust gases are expelled, the momentum changes. The new velocity of the rocket is the original velocity plus the change in velocity, (v + dv), the mass of the rocket is M minus the change in mass of the rocket, (M - - dM) or (M + dM), and the momentum of the exhaust gasses is becomes the change in mass of the rocket, -dM times the exhaust velocity, U.

Mv = -dM U + (M + dM)(v + dv)

=

The total momentum of the system at time (t) is simply M times v. If the rocket isn’t moving in this frame of reference, then Pi = 0

The total momentum in the system at some time later (t + dt). The momentum of the exhaust is

-dM U. The momentum of the rocket is (M + dM)(v + dv).

If we let u equal the speed of the exhaust relative to the rocket, then u = (v + dv) – U or U = v + dv – u.

Substituting into the equation above gives –dM u = M dv or in other words, the change in mass (the mass of the exhaust material) times the exhaust velocity = the total mass of the rocket times the change in velocity. Dividing both sides by dt gives:

dM/dt u = M dv/dt

dM/dt is the rate of change in mass of the rocket, or fuel consumption. u is the exhaust velocity. M is the mass of the rocket, and dv/dt is the acceleration of the rocket.

So, a = Ru/M

This equation appears to apply best to a liquid fuel type motor where the pressure in the reaction chamber and the resulting exhaust velocity stays relatively constant. Since the solid rocket motors have a varying pressure during the burn, the exhaust velocity changes, and the acceleration isn’t constant, I don’t think this equation would be this simple.

Finding velocity from dv = - u dM/M and applying the integral gives

Vf + Vf = u ln Mi/ Mf