This question already has an answer here:
Sorry to repeat this question, but I got no definite answer last time, so I am trying to state the question more clearly.
Power = force x distance / time = force x velocity.
So if we apply a small constant force, as velocity increases, we get power which keeps increasing with the velocity. How is it possible to obtain increasing power from a small constant force?
Specifically, consider a rocket in space, which has a large supply of fuel, burns fuel at a small constant rate, meaning the input power is constant and the generated force (thrust) is also constant. As it accelerates from zero to millions of m/s, we are getting increasing power from the small constant force. How is this possible? Where does the increasing power come from, when the power from the fuel was constant?
- We do not approach the speed of light, so I am not seeking a relativistic solution.
- I am not considering the reduction of mass of the rocket as it burns fuel. If we considered the reduced mass, then it would increase the obtained acceleration, velocity and power still further, from the small constant input power, so would not solve the problem.
- Rocket theory states that thrust does not depend on the velocity of the rocket with respect to the observer, it only depends on the exhaust velocity relative to the rocket, so as the rocket accelerates (to beyond the exhaust velocity), the thrust really will be constant, by burning a small constant amount of fuel.
Previous attempts at solution
The answer which came closest to solving the problem was that when the rocket is at rest, we are accelerating both the rocket and exhaust in opposite directions, so we are spending the energy of the fuel on both, and only a small portion of the input power is used for accelerating the rocket, most of it is used for accelerating the exhaust. As the rocket speeds up, the velocity of the exhaust relative to the observer reduces, so we spend more of the input power in accelerating the rocket and less on accelerating the exhaust. However, this answer fails as the rocket exceeds the exhaust velocity, and both the rocket as well as the exhaust are moving forward relative to the observer. Now, as the rocket accelerates forward, the exhaust accelerates forward too, so the power for accelerating both keeps increasing again, so we get an increasing power from a small constant input power. How is this possible?
Also someone commented that we are not getting increasing power, but we have to provide increasing power to keep accelerating under a small constant force. How does this help to solve the problem in our scenario? We are still burning fuel at a small constant rate and the rocket keeps accelerating. Where does the extra power that we need to “provide” come from? Why does the rocket keep accelerating although the equation clearly states that it cannot do so unless we provide increasing input power (increasing fuel)?