At the same starting radius, how does speed change an orbit?
Predict the path below and above circular speed.
- Pause and set speed to 0.8 times circular speed.
- Run, then reset and compare ratios 1.0 and 1.2.
- Record closest and farthest distances.
What to measure
Compare radius r, speed v, elapsed time and energy error.
The idea underneath
Circular motion requires gravity to provide exactly the inward acceleration needed. Other launch speeds produce eccentric paths, collisions or escape, depending on energy and the planet’s radius.
v_c = √(μ/r); T = 2π√(r³/μ)μ=GM in m³/s²; centre distance r in m; v_c in m/s; T in s.
Build the argument
- Start from F = GMm/r² and divide by satellite mass: a = μ/r².
- For circular motion set v²/r = μ/r². Then v = √(μ/r) and T = 2π√(r³/μ).
- Combine kinetic and potential energy per mass: ε = v²/2 − μ/r. Its sign separates bound and unbound trajectories.
Worked example 1
With μ=3.986×10¹⁴ m³/s² and r=7×10⁶ m, v_c=7546 m/s.
Worked example 2
At that radius T=5829 s≈97.1 min. Doubling r multiplies T by 2√2.
Applications
Orbital mechanics connects satellite navigation, communications and interplanetary transfers. A launch is a choice of position and velocity, not a command to follow a circle.
Common misconception
Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.
A further investigation
What radius factor doubles a circular orbital period?
Check your understanding
Where does acceleration point?
- Toward the centre
- Along velocity
- It is zero
Explanation
Constant speed can still mean changing direction. The challenge answer is 2^(2/3)≈1.587.
Model limits
Newtonian two-body model with fixed central mass. No drag, third bodies, oblateness or relativity. Visual sizes are illustrative; initial conditions are fictional, not live ephemerides.
University Physics Volume 1, 13.4 Satellite Orbits and Energy