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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    $\begingroup$ The rocket is expelling a lot of stuff downward. $\endgroup$ – ACuriousMind Nov 24 '15 at 12:59
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    $\begingroup$ Energy conservation means total energy is constant. In this case there are other types of energy involved (in addition to KE and PE) which you need to consider in your energy balance. $\endgroup$ – pentane Nov 24 '15 at 13:38
  • $\begingroup$ OK let's theoretically assume that any other dissipative factors r absent only gravitation is present... $\endgroup$ – Raj Nov 24 '15 at 14:10
  • $\begingroup$ ** note that here fuel being a part of rocket it can not be considered as no external agent force... $\endgroup$ – Raj Nov 24 '15 at 14:13
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    $\begingroup$ Welcome to physics.stackexchange, Raj! Please don't be discouraged by the downvotes on your first question. However, for questions like the one you asked you might get help faster if you first discuss them with your physics instructor or with a friend. $\endgroup$ – Martin J.H. Nov 24 '15 at 14:16

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:

Hint: The answer to both questions involves the exhaust gases.

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  • $\begingroup$ Hey Martin all u spoke r true about rocket fuel & momentum conservation. But my prime problem is about validity of conservation of energy IN PRESENCE OF GRAVITY !! i.e. K.E+ P.E must always be constan( which is of course happening in some way)... along with momentum.... But I can't find any satis factory explanation to that $\endgroup$ – Raj Nov 24 '15 at 15:34
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    $\begingroup$ @Raj Your view of the law of conservation of energy is too limited, and is incorrect. It does not say that KE + PE is constant. It says that total energy is constant. You must include the chemical energy stored in the fuel. $\endgroup$ – garyp Nov 24 '15 at 16:22
  • $\begingroup$ As the rocket moves upward, The chemical ignition does not ceases until it is about to end the rate of chemical reaction rmains nearly constan t. $\endgroup$ – Raj Nov 24 '15 at 17:10
  • $\begingroup$ @Raj - you have asked a question, and gotten a decent answer. Stop holding on to the (incorrect) concept that you asked the question about in the first place. Clearly something bothered you enough to ask the question, and those concerns have been answered. $\endgroup$ – Jon Custer Nov 24 '15 at 18:34

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