Optimizing Satellite Orbits?

📌 CONCEPT: A conic section is a curve obtained by intersecting a cone with a plane, which can be a circle, ellipse, parabola, or hyperbola, and is used to model the orbit of the satellite.

📐 RULE / FORMULA: To find the closest approach of the satellite to the Earth's surface, we can use the formula for the distance of the satellite from the Earth's center, which is given by the equation for an ellipse: d = a(1 - e), where d is the distance from the Earth's center, a is the semi-major axis, and e is the eccentricity.

💡 WORKED EXAMPLE: Let's say the satellite's maximum distance from the Earth's surface is 36,000 km and the minimum distance is 35,786 km. We know that the maximum distance is equal to a + 35,786 km, and the minimum distance is equal to a - a(1 - e), where a is the semi-major axis. Solving for e, we get e = (a - 35,786 km) / a. Given that the maximum distance is 42,164 km, we can solve for a and then find the minimum distance.

⚠️ COMMON MISTAKE: Students often forget to convert the units of distance to the same unit, which can lead to errors in calculations.