Consider a rocket moves upward with some acceleration for a very small time say '$dt$' then the kinetic energy increases (for acceleration). as well as as the potential energy increases (due to height increase).
How is then $K.E +P.E=$constant?
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Let me discuss a simpler version of your rocket-question: one where there is no gravity, so that we don't have to worry about gravitational potential energy.
Consider a rocket in free space (vacuum), and consider that the rocket is at rest. Now the rocket fires it's engine for a short time. The engine accelerates the rocket. The rocket now has kinetic energy (and momentum, but that's another question). Where does the energy come from?
The energy comes out of the rocket fuel. For a very simple rocket (simple for physicists, not simple for engineers!), you could use a rocket that has two tanks: A hydrogen tank, and an oxygen tank. When you want to fire the rocket engine, you mix the hydrogen and oxygen in a chamber and ignite the mixture. The mixture reacts to water:
$$2\,\text{H}_2 + \text{O}_2 \quad \longrightarrow \quad 2\, \text{H}_2 \text{O} $$
This is an exothermic reaction, so the "exhaust gas" ($ \text{H}_2 \text{O}$, or water vapor) is hotter than the rocket fuel. You send the hot exhaust gas out of the rocket nozzle. The hot exhaust gas goes one way, and the rocket is pushed the other way.
The total energy is conserved: The exhaust gas (water vapor) has a lower chemical energy than the rocket fuel. The rocket has a higher kinetic energy. (The exhaust gases also have a higher kinetic energy!) But the total energy (kinetic energy + chemical energy) is conserved.
It works roughly the same way with a rocket that is fired near a planet, only that you also have to consider the gravitational potential. But the idea is always the same: The kinetic energy (and the potential energy) comes out of the rocket fuel. The energy conservation law in this case can be stated as
$$ \text{K.E.} + \text{P.E.} + \text{C.E.} = \text{const.} \quad, $$
whith $\text{C.E.}$ as the chemical energy (which really is just another form of potential energy).
Note that the formula that I gave for the energy conservation is still a gross simplification. The law of energy conservation simply states that the total energy is conserved. If you want to write "energy conservation" as a formula, you have to make sure that you include all terms relevant to the model that you use. In the rocket model, this means you have to include the chemical energy.
Follow-up questions to think about are:
- Why is the momentum conserved when the rocket starts to move?
- Why does the kinetic energy of the rocket increase by a larger amount, when the rocket engine is fired again on a moving rocket?
Hint: The answer to both questions involves the exhaust gases.
Martin J.H.
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