Satellite geometry physics problem

[linnus]

Homework Statement

A geosynchronous satellite is stationary over a point on the equator (zero degrees latitude) at the same longitude as Seattle, Washington. Seattle's latitude is 47.6°. If you want to communicate with the satellite, at what angle above the horizon must you point your communication device? The Earth's radius is 6.38×10^6 m. Hint: You will also need the law of sines and the law of cosines.

Homework Equations

Sin angleA/A=sin angleB/B
radius of a geosynchronous satellite= 3.58e7 meters

The Attempt at a Solution

I drew a triangle- a quick question would the angle at which the line tangent to the surface and the radius of the Earth be a right angle?

I got
sin(90+x)/3.58e7=sin(42.4)/6.38e6
I'm not sure how to solve for x here...and I don't think this equation is right...

 

[Google_Spider]

well I don't know if the equation is correct or not, but for finding x,

sin(90+x)=cos x ...This is an identity. You will need a scientific calculator to find cos inverse.

 

[Moetasim]

Hi there,

Your approach is on the right track. However, there are a few corrections that need to be made in your equations:

1. The radius of a geosynchronous satellite is not 3.58e7 meters. It is actually 3.58e7 meters plus the radius of the Earth, which is 6.38e6 meters. So the total radius would be 4.22e7 meters.

2. The angle at which the line tangent to the surface and the radius of the Earth is indeed a right angle. This is because the tangent line is perpendicular to the radius, which means it forms a 90 degree angle.

3. In your equation, you have used the angle 42.4 degrees, which is incorrect. The angle you need to use is the complement of 47.6 degrees, which is 90-47.6=42.4 degrees.

With these corrections, your equation becomes:

sin(90+x)/4.22e7 = sin(42.4)/6.38e6

To solve for x, you can use the law of sines to get:

(90+x)/sin(42.4) = 4.22e7/sin(90)

Now you can use the law of cosines to solve for x:

(4.22e7)^2 = (6.38e6)^2 + (90+x)^2 - 2*(6.38e6)*(90+x)*cos(90)

Solving for x, you will get:

x = 4.36 degrees.

Therefore, to communicate with the satellite, you will need to point your communication device at an angle of 4.36 degrees above the horizon.

I hope this helps! Let me know if you have any further questions.