This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To estimate the speed and period of a satellite close to the Earth's surface, we use the given radius of orbit and standard physical constants. Given: • Radius of orbit, r = 6.4 × 10^6 m (assumed to be equal to Earth's radius) • Acceleration due to gravity at Earth's surface, g = 9.8 m/s^2 (a) Estimate the speed of the satellite. Step 1: Use the formula for orbital speed for a satellite orbiting close to the Earth's surface. The orbital speed v is given by: v = sqrt(gr) Substitute the given values: v = sqrt((9.8 m/s)^2) × (6.4 × 10^6 m) v = sqrt(62.72 × 10^6 m)^2/s^2 v = sqrt(6.272 × 10^7 m)^2/s^2 v ≈ 7919.6 m/s Rounding to two significant figures: v ≈ 7900 m/s The speed of the satellite is 7900 m/s. (b) Estimate the period of the satellite. Step 2: Use the formula for the period of orbit. The period T is the time taken for one complete orbit, given by: T = (2 r)/(v) Substitute the values for r and the calculated v: T = 2 (6.4 × 10^6 m)7919.6 m/s T = 4.0212 × 10^7 m7919.6 m/s T ≈ 5077.7 s Rounding to two significant figures: T ≈ 5100 s To convert to minutes: T = 5077.7 s × 1 min60 s T ≈ 84.6 min Rounding to two significant figures: T ≈ 85 min The period of the satellite is 5100 s or 85 min.