# Satellite tracking - Dynamics, Angular velocity, Circular motion

1. Jun 7, 2010

### blueparukia

Been sick for a bit so missed a fair bit of "advanced" maths including locuses, particularly parabolas, but what's worrying me more at the moment is dynamics. So got a moderately difficult question here to help catch me up. I don't really comprehend any of the rules and this question is tricky because it seems I'm gonna need trig (albeit basic trig) to find distances and stuff as well as dealing with time.

1. The problem statement, all variables and given/known data

A satellite traveliling in a circular orbit 1600km above the Earth is due to pass over a satellite tracking station at noon. Assume that the satellite takes two hours to make one complete orbit and that the radius of the Earth is 6400km.

If the tracking antenna is aimed 30 degrees above the horizon (an imaginary line, tangent to the Earth and passing through the station), at what time, to the nearest second, will the satellite pass through the beam of the antenna?

2. Relevant equations
w = dx/dt (differential of x/theta n respect to time)
v = wr
T (period) = 2pi/w
Theta = wt

Measure velocity in radians per hour

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2. Jun 7, 2010

### Coto

Since we can find the speed of the satellite (given the known quantities in the problem), it suffices to find the distance between the point directly above the tracking station and the point where the satellite passes through the tracking beam.

What formula would you use to find this distance? So what do we need?

3. Jun 8, 2010

### blueparukia

Yeah see that's kinda where I'm having the problem. I think I've worked out the velocity, and I could use trig to find out a side for an approximate distance, but not on the curve. I'm fairly sure I'm trying to find out how many radians it has covered rather than metres but I'm not sure.

4. Jun 8, 2010

### Coto

Well, how do you find out the arc length of a circle? (Hint: http://en.wikipedia.org/wiki/Arc_(geometry))

So what angle do you need in order to find the distance the satellite will travel between those two points?

5. Jun 8, 2010

### blueparukia

Arc length is not part of the syllabus and I'd get marked or not rated trying that on a test :p

The angle I would need to find is of course the one at the centre of the Earth. But now this is just getting back to trig, so I should be good. If the angle between the horizon and the tracking station beam is 30 degrees (pi/6) than the angle between the tracking beam and the position at noon would be pi/3.

However the middle triangle is not a right angle triangle whereas the triangle to the center of the earth is. I'm sure this can be done without the sin/cosine rules and there's something in my mind saying that the angle I'm after is pi/6, but I'm not sure - been a year or two since I've done trig with multiple triangles.

6. Jun 8, 2010

### Coto

Ok.

Still, to solve the problem you will need to find the angle at the center of the earth. As you mentioned, it is just trig. I suggest looking up the sine law (http://en.wikipedia.org/wiki/Law_of_sines).

Good luck!

7. Jun 8, 2010

### Coto

Ah yes. Once you have found the angle at the center of the earth (in radians), your suspicions are correct and this is the right idea!

8. Jun 8, 2010

### blueparukia

Cheers, is using the sine rule necessary? I know it well enough, but something tells me this can be solved by simple ratios, but we'll see :)

Cheers

9. Jun 8, 2010

### Coto

It seemed to me to be the easiest way. Like any trig/geometry problem, there are many different routes to get to the same solution.

10. Jun 8, 2010

### Susanne217

You remember the Universal law of of gravity which originates from Newton 2

$$F = m_{sat} \cdot a_c = G \cdot \frac{m_{earth} \cdot m_{sat}}{r^2}$$

$$a_{c} = \frac{G \cdot m_{earth}}{r^2} = \frac{v^2}{r}$$

Thus $$v = \sqrt{\frac{G \cdot m_{earth}}{r}}$$

How is that?

Last edited: Jun 8, 2010
11. Jun 8, 2010

### blueparukia

Yeah I think this is going a little too far :p This isn't actually a real world situation, just an excercise, that is only allowed to use the formulas I mentioned above plus trigonometry. Thanks for all your help though, it's been most informative.

12. Jun 8, 2010

### Coto

Overkill :). And this only finds a velocity which can be found using the data of the problem in a more direct manner.

13. Jun 8, 2010

### blueparukia

Hmm, I've just spent a good 45 minutes trying to find the angle with no luck. So it turns out I'm alright with the new angular velocity, I just can't do simple trig.

There's no way I can conceive this (sin/cosine rule possibly, but I don't understand them enough to be able to make use of them), so I tried some trig functions but they all require a side or angle I don't have, so I've tried doing the sum of all angles = 180 degrees, but again the information seems insufficient to use that to find any extra angles.

Some help would be appreciated thanks :D

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14. Jun 8, 2010

### rock.freak667

If you call the angle between the long orange line and the long red line as θ, and you know that the angle between the radius of the Earth and the imaginary line is 90°, what is the other angle in the triangle?

The you will have two sides and two angles. Sine rule should work to get the angle θ.

15. Jun 8, 2010

### blueparukia

No idea, that's what I was trying to work out with sum of all angles = 180 degrees. And there's no other trig I can think of that will solve it without knowing theta or the other lengths.

16. Jun 8, 2010

### rock.freak667

I'll redraw what I meant:

http://img291.imageshack.us/img291/8250/newaj.png [Broken]

Since the object is traveling in a circle, shouldn't the effective radius i.e. the long red and orange lines be the same?

Last edited by a moderator: May 4, 2017
17. Jun 8, 2010

### blueparukia

Huh, common sense. It strikes again. Thanks a lot.

18. Jun 8, 2010

### blueparukia

OK, I can't figure this out. I'm stumped, I'm frustrated, I can't answer the question, it's been 4 days and the trig is just standing in the way of what I need to know. I am not skilled enough at trig to be able to work this out. The sides can't be equal on a right angle triangle, so if I connect all the dots I get an isoscoles triangle with insufficient data to complete the problem. All I want is the angle, I do not remember anything about trig about from the basic ratios and I feel this answer attempt has been made overally complicated.

At any rate, diagram attached. I want this question answered today and any help given will be hugely appreciated. Thank you all.

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19. Jun 8, 2010

### rock.freak667

http://img718.imageshack.us/img718/6299/54193918.png [Broken]

I removed the confusing parts of your diagram. You are only dealing with triangle ABC, you have angle A=θ, angle B is shown (90+30 gives?) and you can write angle C in terms of θ, since you know the sum of the interior angles of a triangle = 180°.

Then just apply the sine rule.

Last edited by a moderator: May 4, 2017
20. Jun 9, 2010

### blueparukia

Yes I think I got it! Thank you so much, I have an answer :D Thank you very, very much. Hoping the answer is ~40 degrees.