Work and energy balances
Putting a satellite into orbit is not just "lifting it up": you also have to give it the right speed so that it does not fall back down. This page connects those two energy costs using a single language, that of work and energy, and helps you understand real paradoxes such as why a satellite that slows down because of air drag ends up orbiting faster.
Work done by the gravitational field
If you throw a baseball straight up, gravity slows it down until it stops; as it falls back, gravity speeds it up without any other force being involved. How do we quantify the energy that the gravitational field exchanges with an object as it moves away from or towards the planet?
A probe moves away from the Earth in a radial direction. a) Draw in your notebook the gravitational force on the probe and its displacement, and reason what sign the…
A probe moves away from the Earth in a radial direction.
- a) Draw in your notebook the gravitational force on the probe and its displacement, and reason what sign the work done by the gravitational field has during the motion away.
- b) Write down the relationship between that work and the change in potential energy, $W_{campo} = -\Delta E_p$, and explain its meaning.
You can do it on paper and send a photo with your answer.
Zero work in a circular orbit
A satellite describes a stable circular orbit around the Earth. Look at the diagram: a) Given that the gravitational force is perpendicular to the velocity at every…
A satellite describes a stable circular orbit around the Earth. Look at the diagram:
- a) Given that the gravitational force is perpendicular to the velocity at every point of the orbit, what is the work done by the gravitational field in one complete orbit?
- b) Do the kinetic energy, potential energy, and mechanical energy of the satellite change along that orbit? Justify your answer.
Putting a satellite into orbit: two different energy costs
A suborbital rocket reaches a height of 100 km but immediately falls back to the ground, whereas the International Space Station orbits at about 400 km for decades without falling. Why is going up not enough to stay in space, and what extra energy cost is needed to sustain the orbit?
To place a satellite into a certain orbit, its potential energy needs to be increased by $\Delta E_p = 5\times10^{9}\ \text{J}$ and, in addition, it must be given an…
To place a satellite into a certain orbit, its potential energy needs to be increased by $\Delta E_p = 5\times10^{9}\ \text{J}$ and, in addition, it must be given an orbital kinetic energy of $E_c = 4\times10^{9}\ \text{J}$.
- a) Calculate the total energy that must be supplied by the launch vehicle.
- b) Explain why putting a satellite into orbit does not consist merely of "lifting" it up to that height.
Changing orbit: the satellite paradox
Balances in falls, orbital decay and interplanetary journeys
NASA tracking shows that satellites in low orbit, as they brush against the thin upper atmosphere and sink toward denser layers, complete each revolution in less time. How can a resistive, dissipative force make a body end up moving faster?
Back to the beginning: A satellite that slows down because of drag ends up going faster because, as it loses height, the gravitational field does positive work that is twice the energy dissipated by drag ( $W_{field} = 2\,|W_{drag}|$ ), so half of the drop in potential energy is converted into kinetic energy and its orbital speed increases ($v = \sqrt{G M/r}$).