Calculating Earth's Distance from the Sun Throughout the Day

[Lizzynagel09]

Homework Statement

As Earth rotates on its axis, a point on the equator gets closer to and farther from the Sun. Ssume that the center of the Earth is 93 million miles from the Sun and that the radius of Earth is 4 thousand miles. The period of Earth's rotation is, of course, 24 hours.

Assuming that the distance varies sinusoidally with time, wrtie an equation for the distance as a function of time of day, letting t=0 hours represent midnight.

Homework Equations

none

The Attempt at a Solution

A=6
B=Pi/12
C=0
D=1
so: 6cosPi/12(x-1)

is this right?no?
B=

 

[Lizzynagel09]

help me dick!

 

[EnumaElish]

Think of a sinusoidal function, doesn't it look like a repeated half-circle? So you know the shape; all you have to do is to "fit" this shape to represent Earth's rotation.

It'd be easier to work in million units: write 93,000,000 simply as 93. Say at t=0 (midnight), distance of point E on Earth to sun is given by f(0) = 93. How many hours does it take to get back to f(T) = 93? That is, do you know what T is? (You do, it's a given.) That tells you the horizontal "stretch" of your sinusoidal function relative to the x axis.

Now fix the height of your function relative to the y axis: what is the farthest distance that any point on Earth can be away from the sun? (Simple addition.) And, at what time is point E at that distance? Do you see that the answer is T/2?

How many degrees has the Earth rotated in T/2 hours? Your sinusoidal function then has the form sin(0) + C = sin(k*pi*T) + C = 93; and sin(k*pi*T/2) = farthest distance from the sun, where k is a normalizing constant. Solve first for C and then for k.

 

[Lizzynagel09]

idk what you mean by E. i am still confused. why did you change it to sin? what is k?

 

[EnumaElish]

Think of the Earth as a rotating circle, and the sun as a point (S). Mark any point on the circle as point E. When you connect S to the center of the circle, the distance is 93.

Since the function is sinusoid, I thought the problem is asking for an exact (analytic) solution, which would be the sine function. k is the constant coefficient that you need to solve from sin(k*pi*T/2) + C = farthest distance between E and the sun, after you solve for C from sin(0) + C = 93 (which is easier).