Problem 39 - Contemporary Precalculus

Question

Assume that the earth is a sphere of radius 3960 miles. A satellite travels in a circular orbit around the earth, 900 miles above the equator, making one full orbit every 6 hours. If it passes directly over a tracking station at 2 P.M., what is the distance from the satellite to the tracking station at 2: 05 P.M.?

Step-by-Step Solution

Answer

Answer: 4009.76 miles

Step 1: Find the radius of the satellite's orbit

To find the radius of the satellite's orbit, add the radius of the Earth (3960 miles) with the height of the orbit above the Earth (900 miles). This gives us the radius of the satellite's orbit (Rs): Rs = 3960 + 900 = 4860 miles

Step 2: Calculate the angular velocity of the satellite

To find the angular velocity (ω) of the satellite, use the fact that it completes one full orbit (2π radians) in 6 hours (360 minutes): ω = (2π radians) / (360 minutes)

Step 3: Determine the angle the satellite moves in 5 minutes

We know that the satellite passes directly over the tracking station at 2 P.M., which means it's 0 degrees from the station at that time. Now we want to find the angle it moves in 5 minutes. To calculate this angle (θ), multiply the angular velocity (ω) by 5 minutes: θ = (2π radians / 360 minutes) * 5 minutes θ = π/36 radians

Step 4: Calculate the horizontal distance traveled by the satellite

As the satellite revolves around the Earth, it moves horizontally away from the tracking station. To calculate this horizontal distance (HD), use the equation for arc length: HD = Rs * θ = 4860 miles * π/36 radians ≈ 420.11 miles

Step 5: Calculate the distance from the satellite to the tracking station

Now we have a right triangle, where the Earth's radius (3960 miles) is one leg, the horizontal distance traveled by the satellite (420.11 miles) is the other leg, and the distance between the satellite and the tracking station (d) forms the hypotenuse. Utilize the Pythagorean theorem to find the distance d: d^2 = (3960 miles)^2 + (420.11 miles)^2 d = √((3960 miles)^2 + (420.11 miles)^2) ≈ 4009.76 miles Thus, at 2:05 P.M., the distance from the satellite to the tracking station is approximately 4009.76 miles.

Key Concepts

Angular Velocity

Imagine watching a majestic eagle gliding in circles above; the speed at which it completes these circles is similar to what we call the angular velocity in physics. It's a measure of how fast an object moves through an angle in a specific time—just like how fast the eagle covers a section of the sky. For objects moving along a circular path, such as a satellite orbiting the Earth, angular velocity represents how many radians (the angular measurement) it goes through per unit of time. In our satellite's case, it sweeps out 2π radians (or one full circle) every 6 hours. Hence, to determine its angular speed, we divide 2π radians by 360 minutes to get the velocity in radians per minute. This concept is crucial for tracking movements and predicting future positions of objects in circular motion.

Arc Length

Now, let's connect the dots between the swift turn of a roller coaster and our orbiting satellite. The distance along the curve of the coaster that leaves riders breathless is akin to what we call the arc length in the trajectory of our satellite. Arc length is simply the distance along a circular path. Think of it as measuring a piece of string along the edge of a circle. The formula to find this length is the product of the radius of the circle and the angle it spans in radians. By applying this formula to our satellite, we calculate how 'far' it travels along its orbit in 5 minutes. This is vital for understanding the satellite's position relative to different points on Earth, such as the tracking station.

Pythagorean Theorem

Stepping back in time to ancient Greece, we can thank Pythagoras for one of the fundamental principles of geometry: the Pythagorean theorem. It's a seemingly magical formula that connects the sides of a right-angled triangle. Specifically, the square of the hypotenuse (the longest side opposite the right angle) is equal to the sum of the squares of the other two sides. In our satellite scenario, we create a right-angled triangle by using the Earth's radius and the arc length that the satellite has traveled. The missing piece of this puzzle is the straight-line distance from the satellite to the tracking station, and the Pythagorean theorem helps us find this final side, ensuring a precise and calculated approach to understanding spatial relationships.

Circular Orbit

Think of a ring around the planet; just like Saturn's rings, our satellite maintains a circular orbit around Earth. A circular orbit is a path that keeps a constant distance from the center body and is shaped like a circle. Maintaining this specific path requires careful calculation of speed and direction, ensuring the satellite does not spiral away into space or fall back to Earth. In relation to our problem, the satellite's orbit at 900 miles above the equator defines a circle with the center at Earth's core, and knowing this orbit's radius helps us navigate the geometry needed for satellite tracking.

Satellite Tracking

Just like tracking your favorite running route using a GPS watch, scientists and engineers monitor satellites using satellite tracking. This involves calculating positions, predicting future locations, and understanding the satellite's current and future proximity to various points on the Earth. Essential to this tracking are the concepts of angular velocity and arc length, as they help determine the satellite's path and speed. In our example, we've used these fundamental concepts to find the exact distance between a satellite and a tracking station at a specific time, illustrating how mathematics gives the power to track objects far above us with remarkable precision.