# The Rocket Equation

7
In deriving the rocket equation, there is one part I don't understand. The velocity of exhaust with respect to the body is assumed to be constant, where:

v(exhaust wrt body)=v(exhaust wrt inertial)-v(body wrt inertial)

So assuming a constant mass flow rate, the rocket propellant exerts a constant force on the rocket and hence in space, uniform acceleration. But how can an observer on the accelerating rocket observe the rocket propellant being ejected with a constant velocity?

2. ### turin

2,326
This is not correct. The mass of the rocket decreases at a rate equal to the exhaust mass rate. The constant exhaust velocity should be w.r.t the rocket.

7
Oops. So assuming a constant mass flow rate the propellant exerts a constant force on the rocket so the rocket's acceleration increases as follows:

a(t)=F[1/m(t)] where F is a constant (until fuel runs out) where m(t) is the mass of the rocket at time t.

m(t)=m(initial)-bt where b is a constant (mass flow rate)

Hence v_rocket(t)=-Fln(m(t))/b assuming v(0)=0

But why would an observer in the rocket observe a constant propellant velocity?

4. ### turin

2,326
Because the propelling mechanism is in the rocket.

7
Oh, yeah. The engine is on the back of the rocket, so it accelerates with the rocket, so if an engine ejects exhaust at a velocity v, then this is what is observed from the rocket's point of view.

Now I feel somewhat embarrassed, but at least the rocket equation makes sense now.

6. ### turin

2,326
You should not. No one knows everything. You should only feel embarrassed if you refuse to ask a question out of fear of sounding stupid. Don't let your transient embarrasment prevent your permanent understanding:
Have fun learning.