Quote:
Originally Posted by PairTheBoard
I thought that would be the explanation. But I'm still puzzled by this. Suppose you're in a rocket ship floating free in space. Suppose you have a propellant which when burning at the constant rate of p-units/sec applies a constant force F on the ship thereby accelerating the ship at a constant acceleration A (assume the propellant has negligent mass compared to the ship's mass). It it takes the time T seconds to accelerate the ship from 0 to 1 mile/hr then it should also the same time T to accelerate the ship form 1 to 2 mile/hr. Then the energy required measured in units of propellant should be the same pT in both cases. Yet the second case adds more kinetic energy to the ship than the first case. What's wrong with this picture?
PairTheBoard
One needs to be thorough about this. Basically its correct that from the perspective of of constant fuel used the rate of energy consumption is constant or nearly so or can be made so easily. But how much goes to the ship?
What is happening there in a spaceship is the conversion of chemical or nuclear energy to kinetic energy of the exhaust particles or ions or photons ultimately in order to gain momentum in the opposite direction.
You may think the rate is constant but the delivery of energy to the rocket system is not constant if you follow the evolution of the process say like in the rocket equation.
You need to account for the kinetic energy of the fuel also that is non trivial if it is to affect non trivially the momentum of the main ship.
For example over time the kinetic energy of the fuel in the earth system or the laboratory inertial system appears to be changing with time. The fuel initially moves to the left with -v but over time this gets smaller as the rocket gains speed/energy. So the rocket gets a larger part of the released energy than earlier as the exhaust particles get less by coming out as -v+u now. See the difference between 1/2*m*v^2 vs 1/2*m(v-u)^2 as u increases. That implies the rocket system gets more of the share over time.
When the fuel comes out at such huge speeds (eg 4km per second for chemical and 500 km /sec for ions even or more) a little reduction of them has significant impact in energy.
Imagine i send 1/1000 of the rocket out with speed eg v=4000m/sec in the rocket frame (always the same exit speed there) and i gain momentum that means i have now a speed
M*u-m*v=0 => u=1/1000*v. or v= 1000*u
The kinetic energy of the fuel is 1/2*m*v^2 or 1/2*1/1000*M*1000^2*u^2=1000*1/2*M*u^2. Oh wow you may say here. That means the kinetic energy of the fuel is like 1000 times the kinetic energy of the spaceship that is 1000 times bigger in mass but it wont matter in that sense and in fact the opposite.
So the vast majority of energy consumed is going to the fuel in order to get the momentum gain for the ship.
Over time the fuel appears to be coming out with less speed in the laboratory frame as the spaceship gains speed and eventually it may even appear to come out with nothing leaving all the energy to the ship. So the kinetic energy of the fuel declines and the ship's increases in time as expected naturally.
Last edited by masque de Z; 05-12-2018 at 04:45 AM.