Section 2.8: Problem-Solving Strategies

This section introduces systematic strategies for solving gravitation and satellite-related problems efficiently.

Step 1: Identify Known and Unknown Quantities

List all given values and what is being asked. Draw diagrams for satellite orbits and forces if necessary.

Step 2: Select Relevant Equations

Use Newton’s law of gravitation, centripetal force, orbital velocity, and energy conservation equations appropriately.

Step 3: Apply Conceptual Reasoning

Check units, directions, and whether orbits are circular or elliptical. Use approximations if valid.

Step 4: Solve Step-by-Step

Substitute known values carefully. Solve algebraically before plugging numbers. Double-check arithmetic.

Step 5: Verify Results

Check whether the answer is reasonable. Compare with known limits like Earth’s escape velocity, LEO/MEO/Geostationary ranges.

Example 1

A satellite of mass 500 kg orbits Earth at 500 km above the surface. Find its orbital speed and period.

Radius: r = 6371 + 500 = 6871 km = 6.871×10^6 m

Orbital speed: v = √(GM/r) ≈ √(6.674×10^-11 × 5.972×10^24 / 6.871×10^6) ≈ 7610 m/s

Orbital period: T = 2πr/v ≈ 2π×6.871×10^6 / 7610 ≈ 5670 s ≈ 1.58 h

Example 2

Determine the escape velocity from the surface of Mars (mass 6.42×10^23 kg, radius 3.39×10^6 m).

v_esc = √(2GM/R) = √(2 × 6.674×10^-11 × 6.42×10^23 / 3.39×10^6) ≈ 5.03 km/s

Practice Problems

  1. Calculate the orbital speed of a satellite 1000 km above Earth.
  2. Determine the period of a satellite in geostationary orbit.
  3. Compute escape velocity from the Moon (mass 7.35×10^22 kg, radius 1.74×10^6 m).
  4. Find the gravitational potential energy of a 200 kg satellite at 400 km above Earth.
  5. Calculate centripetal acceleration of a GPS satellite.
  6. Determine the speed required for a satellite to maintain a circular orbit around Earth at 2×10^7 m.
  7. Check reasonableness of orbital radius given a period of 90 minutes.
  8. Compare speeds of satellites in LEO vs MEO.
  9. Estimate orbital period of a satellite around Mars at 3×10^6 m above the surface.
  10. Explain why satellites in low Earth orbit experience faster orbital motion than geostationary satellites.
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