Is the energy required to move a massive object equivalent to energy required to move it piece by piece?

Question

Let's say I want to move earth. I can either

• move the whole planet or
• cut it in half, move just the first half and then let the halves gravitate back together.

In the second method, I need less force as I'm just moving one half, but at the same time I need more force to overcome gravity. Also I need some force to stop both halves moving back towards each other so they don't collide violently.

My question is: are both methods equivalent in terms of energy required?

Answer 1

First we should devise a scheme where we end up the same final distance and velocity for either method. This can be achieved by accelerating one half of the cut Earth to twice the velocity we intend accelerating the whole Earth to. This way the Centre of Mass of the two cut halves is moving at the same velocity as the whole Earth and when we stop the rockets that are holding the two cut parts apart, they COM will continue at the same speed as before, due to conservation of momentum and eventually when they come together again, the two rejoined halves will continue with the same speed and Kinetic energy as the whole Earth. Since the final velocities and masses are the same for either method, the energy expenditure in the form of kinetic energy is the same for either method.

Now we have to add the energy used in cutting the Earth in two and the energy used and recovered due to changes in gravitational potential energy separating the two halves over a distance. The potential energy calculations get a bit tricky, because the usual formulas for gravitational force and potential energy assume spherical masses that can be treated as if all the mass is at the centre and that is not the case here for the two cut halves. However we can observe that when we raise a weight above the Earth it requires a certain amount of energy and if use various electro-mechanical devices to recover the energy when we let it fall back to Earth, the energy recovery is unlikely to be 100% efficient. In other words we cant make an energy profit and the best we can hope to do is break even. Therefore the process of separating the two halves and letting them return to each other is almost certainly going to have an energy cost, which puts the two halves idea at a disadvantage.

Now factor in the probably not inconsiderable energy of cutting the Earth in two and it is certain the simplest, safest, most cost efficient and energy efficient method of moving the Earth, is simply to move it without cutting it in two.