Calculating Satellite Speed and Orbit Height

Introduction

• Calculating the speed of a satellite in a circular orbit.
• Understanding centripetal and gravitational forces.
• Deriving equations to find speed, period, and height.

Satellite Speed Calculation

• Centripetal Force: Provided by gravity.

• Gravitational Force Equation: [ F = \frac{G \cdot M \cdot m}{R^2} ]

  • (G): Universal gravitation constant.
  • (M): Mass of Earth.
  • (m): Mass of the satellite.
  • (R): Distance between Earth's and satellite's centers.

• Centripetal Force Equation: [ F = \frac{m \cdot v^2}{R} ]

• Speed Calculation: Derived Equation: [ v = \sqrt{\frac{G \cdot M}{R}} ]

• Distance (R):

  • Earth's radius: (6.38 \times 10^6) meters.
  • Satellite height: 3800 km = (3.8 \times 10^6) meters.
  • (R = 1.018 \times 10^7) meters.

• Calculate Speed:

  • Plug values into the speed equation.
  • Result: (6254.3) m/s.

Calculating the Period of the Satellite

• Using Constant Speed Equation: [ D = v \cdot T ]

  • (D = 2\pi R) (Circumference)

• Period Calculation: [ T = \frac{2\pi R}{v} ]

  • Result: (10,227) seconds.
  • Convert to hours: (2.84) hours.

Geosynchronous Satellite

• Geosynchronous Period: Same as Earth's rotation period.
  • One rotation: 24 hours.
  • Convert to seconds: (86,400) seconds.

Satellite Height Calculation

• Relation between Period and Radius:

  • Derived from speed and gravitational equations.
  • [ R^3 = \frac{G \cdot M \cdot T^2}{4\pi^2} ]
  • [ R = \sqrt[3]{\frac{G \cdot M \cdot T^2}{4\pi^2}} ]

• Calculate Orbital Radius (R):

  • Result: (4.22 \times 10^7) meters.

• Calculate Satellite Height (H): [ H = R - \text{Earth's Radius} ]

  • Result: (3.58 \times 10^7) meters or 35,800 km.

Calculate Satellite Speed in Geosynchronous Orbit

• Speed Calculation using Orbital Radius:
  • Formula: [ v = \sqrt{\frac{G \cdot M}{R}} ]
  • Result: 3072 m/s.

Summary

• Calculated speed, period, and height for satellites in circular and geosynchronous orbits.
• Utilized gravitational dynamics and orbital mechanics.

This comprehensive summary covers how to determine the speed, period, and height of a satellite using equations derived from gravitational and centripetal forces. The explanation includes how these principles apply to both general satellites and geosynchronous satellites.