newtonian mechanics - How is the work done to push a planet over 1m with 1N the same as pushing a feather over 1m with 1N?

Assume there are no other forces acting and the rocket+fuel described do not weigh anything. Also, by rocket I mean engine/thruster, not space shuttle.

Suppose you have a planet, say of mass 1,000,000,000 kg and you push it with a rocket which exerts a force of 1 N. This will accelerate the planet by $10^{-9}\, m/s^2$.

Then, suppose you have a feather of mass 0.005 kg and push it with the same rocket which exerts a force of 1 N, accelerating it by $200\, m/s^2$.

It follows that the feather will traverse a distance of 1 m much faster than the planet and thus spend much less rocket fuel to do so. So, the energy used to propel the planet must be much more than the energy used to propel the feather.

But, work done = force × distance, $1 \times 1 = 1\, J$ for both the planet and the feather. I think I have misunderstood something. How is this possible?

Answer

Let's take a Closer Look at the Whole System: It consists of the Planet / Feather (the accelerated object), the engine / thruster, and (and this is important) particles acellerated by the thruster in direction reverse to the pushing force acting on the planet. Your massless engine, however it works, uses the energy of the massless fuel to accelerate a) The Planet / Feather, and b) some other particles (usualy the burned fuel) in order to satisfy conservation of momentum. The whole Energy of the System is given by: $$ E = E_{planet} + E_{burned fuel} + E_{fuel} $$ The whole momentum is given by $$ P = P_{planet} + P_{burned fuel} $$

To make it short: On your one meter acceleration way, the energy that the planet and the feather get are the same. This isn't true however for momentum: The planet will carry more momentum --> There will be more "burned-fuel-particles" carrying more momentum, and thus also carrying more energy. And that's the energy that is missing in your equation. The whole energy of the system will be conserved.

To look at it more detailed: Force is the time derivative of momentum. The thruster pushing the planet / feather with one 1 N means, that every second there is generated a backward flying "burned-fuel-particle" carrying 1 kg $\frac{m}{s}$ momentum. Since the planet takes (as you noticed) more time to move 1 m, the there will be more particles accellerated backwards. They carry most of the energy that is provided by the fuel.