Conservation of energy in a rocket

[aliens123]

Imagine I have a rocket with a certain amount of energy stored as chemical energy, let's say its 10Js, that exhausts itself after 5 seconds. If I attach this rocket to a (relative to an observers frame) stationary cart in such a way that it pushes the cart, it will add 10J of kinetic energy to the cart system.

But if I attach this rocket to a moving cart, it will still burn for 5 seconds, and because this new cart is moving that means it will apply an equal force for a greater amount of distance. Thus, it should do more work and add more kinetic energy. But didn't it only start with 10J of chemical energy? Or instead of a rocket we could imagine an ideal engine which extract a certain amount of kinetic energy from its environment: A fixed amount of joules. If this engine is moving then it will extract the same amount of energy as heat, but won't it apply an equal force over a now greater distance?

 

[Dale]

You are forgetting to account for the energy in the exhaust. At low speeds in particular most of the energy goes into the exhaust.

 

[Lsos]

Don't discount the kinetic energy of the fuel, which rises as the rocket gains speed and makes up for the apparent deficit in its chemical energy. This is why it's exponentially harder to make faster rockets: because you need more fuel to accelerate the rocket, and then you need more fuel to accelerate the fuel. Luckily, as you discovered, this exponential increase in the rocket's size (and therefore its energy) is not wasted: it gets invested into the rockets's remaining fuel as kinetic energy, which in orbital rockets ends up far exceeding it's chemical energy.

Look up "Oberth Effect"