# Satellite changing distance as θ changes how?

1. Feb 28, 2012

### LearninDaMath

note

Please note: I am not asking for the answer, nor am I asking how to solve a) or b). This question is about what it means that h is changing as θ is changing.

textbook question

An earth observing satellite can see only a portion of the earth's surface. The satellite has horizon sensors that can detect the angle θ shown in the accompanying figure. Let r be the radius of the earth and h the distance of the satellite from the earth's surface.

a) show that h = r(cscθ -1)

b)Using r = 6378km, find the rate at which h is changing with respect to θ when θ = 30 degrees. Express in km/degree

question

The illustration for this question is:

How/why is h changing as the angle changes? Does this imply the earth is traveling in some direction or rotating? Or does it imply the satellite is changing is traveling in some direction...or rotating? It seems like the answer is implying that the satelite getting closer to the earth as the angle changes? Why?

2. Feb 28, 2012

### LearninDaMath

nevermind, i reasoned it out...is it correct, i don't know, but it makes sense to me :)

3. Feb 28, 2012

### Delphi51

θ is the angle on the right hand side of the triangle, at the vertex where the satellite is.
When h is smaller (satellite closer to the Earth), the angle θ must be larger. As h changes, θ must change as well. Draw the diagram with h larger or smaller and you will see that θ has changed with h.

4. Feb 28, 2012

### LearninDaMath

Thanks, that's what I concluded. Appreciate your response.

Would you be able to lend some insight on this question:

It's a classic ladder question from my calc book. But its still physics. I have the correct answer and understand the process of getting to the correct answer.

However, there seems to be no graphical representation of what is going on. I am interested to know how a question like this would be represented graphically.

Since I'm using trig functions, f(θ) = sin(θ) specifically, would I represent the function as a sin graph? And then the derivative would be the line tangent to the sin function at θ = 60?

The derivative of sinθ is cosθ, so the slope of the tan would be cos(60) = 1/2.

However, the actual derivative is 1/2 * 10 since the original function is sin60 = x/10 , so I actually have 10cos60 = 10*1/2 = 5 = Mtan

So the slope of the tangent of sin(θ) at θ = 60 is 1/2 , however, the function in the question is 10sinθ = f(θ) = x ....so is the graph f(θ) = sin(θ) = x or f(θ) = 10sin(θ) = x ...or neither?

5. Feb 29, 2012

### Delphi51

The function is x = 10*sin(θ). You could graph x vs θ and find the slope at θ = 60 to visualize your solution dx/dθ = 10*cos θ = 5.