# If a satellite moves towards Earth, is negative work done against the force of gravity?

A bit confused here - if a satellite approaches Earth, its displacement is in the same direction as the gravitational force (towards the centre of rotation). Therefore, gravity is doing positive work?

I read in my textbook that "negative work is done against the force of gravity" when a satellite approaches Earth. Is it correct to assume that gravity does positive work then?

Is this correct? Conceptually confused right now.

• What other force is present besides gravity? Are we considering drag from the atmosphere? Also keep in mind that if the force has a component in the direction of displacement then the work done by that force is positive. – Aaron Stevens Dec 30 '18 at 4:59
• so, if the satellite approaches Earth and the displacement is towards the centre (like the direction of gravitational attraction), then positive work has been done? – globe1004 Dec 30 '18 at 5:05
• Yes that is the correct application of the definition of work. That's why I'm asking what other forces are supposed to be at play here. You reference some external force, but I'm not sure what that force is supposed to be. – Aaron Stevens Dec 30 '18 at 5:06
• i'm not sure either - my book said 'negative work is done against the force of gravity'. i don't know what does the negative work... – globe1004 Dec 30 '18 at 5:22

The work done by a force on some object is defined as $$\int\mathbf F\cdot\text d\mathbf x$$, where the line integral is over the path the object takes.

Therefore, if the object moves towards the Earth in the direction of the force of gravity, it must be that the force of gravity does positive work.

As you mention the falling object also loses potential energy as a result of this. So another way to see the sign of the work done by gravity is to realize that $$W_{grav}=-\Delta U>0$$

Right now I'm not sure what other force is present that would be doing negative work. Perhaps a drag force due to air resistance as the object falls through the atmosphere?

• thank you! yes, i was confused too since gravitational force is the only significant force. my book did say 'negative work is done against the force of gravity'...not sure what does negative work. likewise, if the satellite moves away from Earth, gravity would be doing negative work? – globe1004 Dec 30 '18 at 5:21
• @globe1004 Yeah your book is using confusing and needles terminology. You can just say a force does positive, negative, or no work. You don't need to say that a force is doing "work against another force". That just makes things confusing. And yes, if the satellite was moving away from Earth then gravity is doing negative work. – Aaron Stevens Dec 30 '18 at 5:23
• @globe1004 Please consider marking an answer as the correct answer for future readers of this question. – Aaron Stevens Dec 30 '18 at 12:46

In this anwsner i will assume that there's only the gravitational force acting on the Satellite.

The satellite is moving towards Earth i.e. the particle moves from $$r+\delta$$ to $$r$$, then $$W=\int_{r+\delta}^{r}\frac{GMm}{r^2} dr = -GMm(\frac{1}{r+\delta}-\frac{1}{r})$$ if $$\delta> 0$$ the integral is positive.

Obs: This is more like an mathematical explanation of Aaron's answner.

If the satellite is on a free falling geodesic the acceleration is force free and the total energy is conserved, so no work is done then. If you accelerate the satellite with a propellant, no matter in what direction, work is done. Then the total energy (which consists of a positive kinetic, a negative potential and a constant rest mass component) increases, either because the positive kinetic part becomes larger than the negative potential, or the other way around.

Edit: I just saw the tag on the question is Newtonian gravity, but what I said is General Relativity. Nevertheless, the part with the conserved orbital energy still holds in Newtonian gravity if the satellite is in free fall.