The Riddle of the Rotating Satellite
A communication satellite orbits the Earth in a circular path. At a certain time, its shadow falls on a particular point on the Earth's surface. After 3 hours, the Earth has rotated 45 degrees, and the satellite has also completed a part of its orbit. If the satellite's shadow is now 1000 km away from the original point, calculate the radius of the satellite's orbit, assuming it remains constant and the Earth is a perfect sphere with a radius of 6400 km.
Use the formula for the arc length in a circle, S=rθ, where S is the arc length, r is the radius, and θ is the angle in radians.
Understanding the Problem:
ReplyDeleteIn 3 hours, the Earth rotates 45 degrees.
The satellite's shadow moves 1000 km on the Earth's surface.
Converting Degrees to Radians:
First, convert 45 degrees to radians because the arc length formula uses radians.
The conversion is done by multiplying degrees with pi/180.
So, 45 degrees in radians is 45 x pi/180.
Using the Arc Length Formula:
The formula for the arc length is S = r theta, where S is the arc length, r is the radius, and theta is the angle in radians.
We know S = 1000 km (the distance the shadow moved) and theta (45 degrees converted to radians).
Calculating the Radius:
Rearrange the formula to solve for r: r = S/theta.
Substitute the values of S and theta into the formula.
By calculating r using the above steps, you will find the radius of the satellite's orbit. Remember to use a calculator for the actual numerical computations, especially for converting degrees to radians and for division.