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Orbit: When traveling through space, you find that you want take a closer look at a planet. You decide that the best idea would be to go into orbit around the planet. You turn to your trusty helms person and tell them to put you into orbit. But a problem with your boosters has come up and the your spaceship can only go 20,000 mph (no faster or slower). The person at the helm looks at you helplessly but you tell them "never fear!" and you run to your physics book:
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1. You know that if the velocity of the body is wrong for orbit, you can adjust radius of the path to make it work. Your steering thrusters still work so all your have to do is make a slight readjustment to your current path (your currently aimed tangent to the planet's edge). You also know that you're 10,000 km from the point on the planet where you would impact if left alone.
2. The first thing that you want to establish is the correct radius of your path. You then start to remember some equations of importance:
Fr = mv^2/r
You then realize that Fr is the force needed to keep you on a circular path. (m is your ship's mass and r is the radius of your intended path) Another equation pops into your mind:
Fg = (GmM)/r^2
You also remember that the gravitational force on your ship will be Fg (as above where M is the mass of the planet)
3. In a stroke of genius you realize that the gravitational pull has to be just enough to keep you on a circular path; they need to be equal!
mv^2/r= (GmM)/r^2
4. Some simple algebra will allow you to find the desired radius:
multiply both sides by r^2: mv^2r= (GmM)
multiply both sides by 1/(mv^2): r = (GmM)/(mv^2)
cancel the ms: r = (GM)/v^2
5. Now that you have the radius you can use trigonometry to determine how much to change your trajectory:
Since you are in the direction tangent to the planet, the radius above (r) minus the radius of the planet (R) is the change in radius of path that you want to induce.
r - R = opposite leg of right triangle
where theta is the change in the angle you need to obtain the proper path radius. (the adjacent leg is the already known 10,000 km distance)
tan (theta) = (r - R)/10,000,000)
or
theta = tan^-1((r - R)/10,000,000)
6. You can clearly see that the last step is to plug in the desired r:
theta = tan^-1( ([(GM) / v^2] - R) / 10,000,000 )
After a few more minutes of converting your velocity to SI, plugging in the values of the particular planet and such your turn to you helms person and say "Twenty three point eight degrees to the left Scotty!" You've saved the day again thanks to your heroic knowledge of physics.
(A little bit of willing suspension of disbelief would lead you to realize that in this case you can disregard the fact that the gravitational pull is constantly increasing your velocity)
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Page By Ben Hoffman