Orbiting Satellite

Task

A satellite orbiting the earth in a circular path stays at a constant altitude of 100 kilometers throughout its orbit. Given that the radius of the earth is 6370 kilometers, find the distance that the satellite travels in completing 70% of one complete orbit.

IM Commentary

This task provides a context for connecting an angle in radians to the arc length intercepted by the angle. There are several approaches: Students can first compute the circumference of a circle of a given radius then take 70% of the answer. Or they can first find the angle the satellite travels through and then use the formula that relates arc length, radius and angle to find the answer.

The task could be used to introduce the formula $\text{arc length} = \text{angle}\cdot\text{radius}.$ This would be accomplished by starting with the first solution, which might be more intuitive and only relies on the formula for the circumference of a circle, and rewriting the computation to bring it into the form of the second solution: $\text{arc length} = 0.7 (2\pi r) = (0.7\cdot 2\pi)\cdot r$.

The task could be used as an in-class activity, practice or assessment, depending how it is situated in a unit.

This task could be extended to interpret the angle as the proportionality constant between arc length and radius (G-C.B.5) by asking for the total distance traveled by satellites that are in orbit at different heights above the earth. For each height the computation would be the same as in solution 2 with only the radius changing but the angle staying constant.

Task based on a problem by Jerry Morris, Sonoma State University. Used with permission.

Solutions

Solution: 1

We can draw a diagram to visualize the situation:

Note that, in the diagram, the inner solid circle denotes the surface of the earth, while the outer dotted circle denotes the path of the satellite in its circular orbit. The radius of the orbital path is $r = 6370\text{ km} + 100\text{ km} = 6470\text{ km}$. Also, since there are $2\pi$ total radians in the central angle of an entire circle, the length of the entire orbital path of the satellite is $$ 6470\text{ km}\cdot 2 \pi \ = \ 12940\pi \text{ km}. $$ Therefore, the distance traveled by the satellite in completing $70\%$ of its orbit is $$ 0.7(12940\pi)\text{ km} \approx 28457 \text{ km}, $$ which is our final answer.

Solution: 2

One full orbit of the satellite is $2\pi$ radians. Therefore, 70% of a full orbit is $2\pi(0.7)=1.4\pi$ radians.

The radius of the orbit is $6370\text{ km}+70\text{ km} = 6470\text{ km}$.

We can use the formula $s=r\theta$, where $r$ is the radius of the circle, $\theta$ is the angle and $s$ is the arc length intercepted by the angle:

$$s=1.4\pi\cdot 6470\text{ km} \approx 28457 \text{ km}.$$