An Interesting Gravitation Problem

AI Thread Summary
The discussion revolves around a gravitation problem involving two satellites launched tangentially at different speeds from a planet. The first satellite achieves a stable circular orbit, while the second, launched at half the speed, requires analysis of its minimum distance from the planet during its trajectory. Participants suggest using conservation of energy and angular momentum to derive the necessary equations for both satellites. The importance of the second satellite's velocity being tangential at its lowest point is emphasized for calculating angular momentum accurately. The problem is ultimately resolved through simultaneous equations derived from these principles.
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Homework Statement


Two satellites are launched at a distance R from a planet of negligible radius. (Yes, that's what the problem says...) Both satellites are launched in the tangential direction. the first satellite launches correctly at a speed v_0 and enters a circular orbit. The second satellite, however, is launched at a speed \frac{1}{2}v_0 . What is the minimum distance between the second satellite and the planet over the course of its orbit?


Homework Equations





The Attempt at a Solution


I thought about using energy. The two satellites both start out with the same potential energy but different kinetic energy.

So satellite one's TME: -\frac{GMm}{R} + \frac{1}{2}mv_0^2

where as satellite two's TME:-\frac{GMm}{R} + \frac{1}{8}mv_0^2

And the second satellite's minimum distance is when its potential is the least...
Somehow I think this problem might relate to angular momentum... L = mvr, but not exactly sure.

THANK YOU =D
 
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You can use the fact the first satellite goes into a circular orbit to figure out M in terms of R and v0 using the centripetal acceleration of an object in circular motion. Let R1 and v1 be the radius at the lowest point. At the lowest point v1 is again tangenential to the planet. So conservation of angular momentum gives you one equation in R1 and v1 and conservation of energy gives you another. Solve them simultaneously.
 
Ah, i c~

and i got it right =D

ThanX
 
Dick said:
You can use the fact the first satellite goes into a circular orbit to figure out M in terms of R and v0 using the centripetal acceleration of an object in circular motion. Let R1 and v1 be the radius at the lowest point. At the lowest point v1 is again tangenential to the planet. So conservation of angular momentum gives you one equation in R1 and v1 and conservation of energy gives you another. Solve them simultaneously.

How can you prove that the lowest point does indeed occur when the velocity is tangent again? And why do we need to know that it is tangent? Is something not conserved otherwise?
 
Last edited:
compwiz3000 said:
How can you prove that the lowest point does indeed occur when the velocity is tangent again? And why do we need to know that it is tangent? Is something not conserved otherwise?

The lowest point occurs when the object is traveling 'horizontally' i.e. perpendicular to the radial vector connecting the object to the planets center. Draw a picture. Knowing the angle between those vectors makes it easy to compute angular momentum. Try it.
 
haha I forgot that angular momentum is a cross product
 
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