Student Resources
Integrals: Energy to Orbit
This diagram estimates the energy required to transport a 1 kg mass from the surface of the Earth to geosynchronous orbit. As shown, the radius of the Earth is 6,371 km and the radius of geosynchronous orbit is 42,164 km.
A Joule (J) is the amount of energy expended to move an object 1 meter (m) with a force of a Newton (N). Proportionally, the energy expended to move an object \(d\) meters with \(F\) Newtons of force is \(E=Fd\) joules. A Megajoule (MJ) is a million Joules.
The graph in the sketch shows the force of Earth's gravity on a 1 kg mass as a function of the distance \(r\) from the center of the earth.
Manipulate the three sliders to answer the following questions.
Use the sketch to read off the first term of \(R_{10}\). What does this number represent in the context of launching a 1 kg payload into orbit?
How far is it (in m) from the surface of the Earth to geosynchronous orbit?
How far is each distance \(\Delta r\) (in m) used in the computation for \(L_{10}\)?
What is the force (in N) used in the computation for the second term in \(L_{10}\)? (Note that the force of Earth's gravity on a 1 kg object at the surface of the Earth is 9.8 N, so your answer should be less than this.)
Use the applet to find \(M_7\) (in MJ). What does this number represent in the context of launching a 1 kg payload into orbit?