# Total work done on a satellite

I have a question about the concept of work in relation to conservative and non-conservative forces. Is Total work done $$=$$ change in kinetic energy or change in total energy $$?$$

This question arose when I was doing a question about the total work needed to move a satellite between orbits. To simplify things, I will use unrealistic values for $$KE$$ and $$GPE$$ but it will hopefully clarify my question.

For example, a satellite has $$GPE = -3$$ $$,$$ $$KE = 5$$ in a lower orbit. It moves to a higher orbit with $$GPE = -1$$ $$,$$ $$KE = 4$$. My understanding is that the satellite does positive work converting fuel to $$KE$$ to instantaneously increase velocity (Hohmann transfer) i.e. raises $$KE$$ to $$6$$. Then as satellite moves off into higher orbit gravity does negative work converting its kinetic energy to potential energy, i.e. increases $$GPE$$ to $$-1$$.

So in this scenario:

Total change in energy $$=$$ $$+1$$

Change in $$GPE$$ $$=$$ $$+2$$

Change in $$KE$$ $$=$$ $$-1$$

In this scenario, what is the total work done? In the question I was doing, the answers stated that the total work done in moving the satellite to a higher orbit was the change in total energy. However, by the work energy theorem the work done should be equal to the change in kinetic energy which is $$-1$$. Also, does the presence of both conservative and non-conservative forces affect the calculation for total work done?

In essence: what is the net work/total work done on the satellite, vs the work done by the satellite vs the work done by gravity.

Usually in problems where you keep track of GPE, you ignore the work done by gravity, as that is taken care of by the GPE.

The total work done by the satellite engine is equal to the change in mechanical energy $$(\Delta KE + \Delta GPE)$$

The total work done by all forces on the satellite is equal to the change in kinetic energy $$(\Delta KE)$$

• Thank you so much for this. I wish my textbooks were as clear as you are. Commented Feb 18, 2021 at 7:02

Confusion can often arise because the system under consideration has not been defined clearly.
Assume $$\rm mass_{\rm Earth} \gg mass_{\rm satellite}$$.

Let the system be the satellite alone and suddenly in some way or other its kinetic energy increases.
In your example the kinetic energy has become $$6$$.
The satellite moves to a higher orbit where the kinetic energy of the satellite is $$4$$ so the change in kinetic energy is $$4-6 = -2$$. The external force which is acting on the system (satellite) is the gravitational attraction of the Earth.
That force does work on the satellite equal to $$-2$$.
The work done is negative because the gravitational forces is in the opposite direction to the displacement of the satellite as it moves to a higher orbit.
Summing up,
$$\text{work done by external forces = change in kinetic energy} \Rightarrow -2 = 4-6=-2$$

Now consider the satellite and the Earth as one system.
Again the change in kinetic energy of the satellite (and Earth) system is $$-2$$ but now the gravitational potential energy of the satellite and Earth system changes by $$-1-(-3) = +2$$ .
The emphasis on the word and is because to define gravitational potential energy there must be a system consisting of at least two masses.
In this case,
$$(\text{potential energy + kinetic energy})_{\rm initial} = (\text{potential energy + kinetic energy})_{\rm final}$$
$$\Rightarrow -3+6 = -1+4 = +2$$

Consideration of the "burn" phase in the time the kinetic energy of the satellite changes from $$5$$ to $$6$$ is more complicated because in effect there is a "super-elastic" collision, like an explosion, during which time chemical potential energy of the fuel is converted to kinetic energy of the satellite and the exhaust gases.
If this occurs over a short interval of time then the linear momentum of the satellite and the burnt fuel can be considered to be conserved as is done when deriving the rocket equation.

Is Total work done = change in kinetic energy or change in total energy?

The energy conservation law always includes all energy. If you from the outside add or remove energy to a system, via heat or work, you must add/remove that in the energy conservation equation as well.

$$\sum E_\text{before}+W+Q=\sum E_\text{after}$$

In order to use this equation, you first must choose a system. Is your system the Earth-and-satellite? Or is it just the satellite?

• If your system is just the satellite, then only kinetic energy is involved within the system (and heat is not relevant): $$KE_\text{before}+W=KE_\text{after}\quad\Leftrightarrow \quad W=\Delta KE$$

• If your system is the Earth-and-satellite, then there is also gravitational potential energy changing within this system: $$KE_\text{before}+GPE_\text{before}+W=KE_\text{after}+GPE_\text{after}\quad\Leftrightarrow \quad W=\Delta KE+\Delta GPE$$

The former, $$W=\Delta KE$$, is what is typically referred to as the work-energy theorem. It is the energy conservation law simplified for a specific scenario. I would not recommend using it, though. Rather, just start from the general energy conservation law every time and you'll avoid confusion of which energies to include.

In this scenario, what is the total work done? In the question I was doing, the answers stated that the total work done in moving the satellite to a higher orbit was the change in total energy.

I think you mean, what is the total external work done. That would be the $$W$$ in the equations above.

And yes, as you can see in the equations above $$W$$ is always equal to the total energy change, not just to the kinetic energy change. Sometimes kinetic energy is the only thing that changes, though, which is why you might have seen that often.

Also, does the presence of both conservative and non-conservative forces affect the calculation for total work done?

It depends on whether the conservative forces act externally or within.

• In the first scenario above with the satellite system, the conservative force of gravity is an external force. The work it does is thus included in $$W$$.
• In the second scenario with the Earth-and-satellite system, gravity is an internal force. So it is not included in $$W$$. Instead, gravity is the cause of GPE being present.

So, as you can see, all forces are always included, but conservative forces in particular can either contribute with work or with potential energy. Whenever you see a GPE term in your equation, then that is the work done by gravity (and then that work is not included in the $$W$$ term).