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Physics Problem Breakdown: Gravitational Potential Energy, Orbits, Escape Velocity

Does talking about gravity, orbits, and planets make your brain want to explode!??

In this video I break down a Physics 1 exam problem that covers gravitational potential energy, orbits, escape velocity, and all the relationships between them that you’ll need to know.

Problem Statement

A space station of mass 400,000 kg, orbits the earth at a height 36000 km above the earth’s surface.

(1) What are the values of the potential energy, and total energy of the space station?

(2) With what velocity must a probe be fired from the space station to escape the gravitational field of the earth? (Re = 6.37*10^6 m; Me = 5.98*10^24 kg)

Solution

Problem from Koofers.com

151119-VProb6-BlogImage1-TJM

Breakdown

151119-VProb6-BlogImage2-TJM

Summary

0:47 – Problem statement breakdown: What we’re asked to do here is, first, find the value of the potential energy and total energy of this space station. Then, find the escape velocity of a probe launched away from that space station.

1:03 – Analyzing the potential energy vs. total energy of the orbiting space station: In the solution we have both an E (total energy) and a U (potential energy) and these two numbers are different (E is greater than U). Why would the two different answers be different and how are we getting those answers?

2:05 – Where the gravitational potential energy equation comes from and why it’s negative: This G*M*M over R equation, comes from the definition of gravity, Newton’s theory of gravitation. If you remember, the definition of work is the force times the distance traveled. Our potential energy is simply this force times the distance away from the other mass. The potential energy ends up negative because the reference point for zero potential energy is an infinite distance away from the other mass. Then as they get closer the potential energy decreases (gets more negative). The reference point for zero potential energy is set up like this because the equation involves the reciprocal 1/r, and it makes the math work out correctly.

4:42 – Where the total energy equation comes from, relating velocity, gravitational force, and centripetal acceleration: This “special” total energy equation is coming from taking the regular equation for total energy (potential energy plus kinetic energy), and then doing a whole bunch algebra to solve for the velocity in the 1/2*m*v^2 part of the equation, and simplifying. This is done by relating the gravitational force imparted on the space station by the earth, and equating it to the centripetal force required to keep it in orbit.

8:02 – Analyzing the escape velocity part of the question: This is a slightly different question because it doesn’t involve an orbit, but it does involve energy. Escape velocity represents the velocity in which we have to launch this probe away from Earth in order to have it “escape” the gravitational pull of Earth. This means at this “escape” point the total energy acting on the probe will be zero. To solve for this velocity we set the total energy equal to zero, and solve (this ends up being the point where the gravitational potential energy imparted by Earth equals to 1/2*m*v^2 of the probe).

Physics StackExchange reference: http://goo.gl/ea6V2t
JHU total energy reference: http://goo.gl/bmslUH
Hyperphysics escape velocity reference: http://goo.gl/hlxKfg

You can also view the video above on Youtube.



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