Think in ratios

Can a dimensionless number reveal an entire family of orbits?

Can a dimensionless number reveal an entire family of orbits?

1. Keep launch ratio at 1 and change mass.
2. Compare characteristic orbital speeds.
3. Predict a time-scale ratio before changing radius.

What to measure

Units and parameter definitions are shown beside every laboratory control and live reading.

The idea underneath

The ratio of launch speed to circular speed removes the units from the launch problem. At fixed relative planet size, equal ratios produce similar classes of trajectory even when mass and radius change. Dimensional analysis also predicts the characteristic time √(r³/μ). Scaling radius without considering the physical planet surface can change collision behaviour, so similarity requires matching every relevant dimensionless ratio.

Build the argument

1. Write a prediction before running the experiment.
2. Hold all but one variable constant, collect repeated measurements, and calculate dimensionless ratios.
3. Compare the observations to an analytic limit and document the remaining discrepancy.

Worked example 1

Doubling μ increases circular speed by √2.

Worked example 2

Doubling r increases orbital time by 2√2.

Applications

A useful model makes a testable prediction and states where it fails. Reproducible settings and exported measurements turn a visual experiment into evidence.

Common misconception

More decimal places do not guarantee greater accuracy. Distinguish discretisation error, uncertainty in inputs, and limits of the physical model.

A further investigation

Which launch ratio marks marginal escape?

Check your understanding

Which launch ratio marks marginal escape?

• √2
• 1
• 2π

Model limits

Zero weight is not zero gravity. Orbital radius is measured from the centre. Numerical energy drift must not be mistaken for real energy production.

University Physics Volume 1, 13.4 Satellite Orbits and Energy