Testing a scaling law: crater diameter and impact energy
The question
How does the diameter of a crater in sand depend on impact energy, and does it follow a power law?
Experiments with balls dropped into sand suggest that crater diameter grows roughly with the fourth root of impact energy. Testing a scaling law with your own data is how real crater science works.
Equipment
- A deep tray of fine dry sand
- Balls of similar size but different masses (for example a steel ball bearing, a glass marble and a hollow plastic ball of the same diameter), and a top-pan balance
- A clamp stand and metre rule
- Callipers or a ruler
- A spreadsheet and eye protection
Method
- For each ball, drop from five heights between 0.10 and 1.20 m, three times each, levelling and loosening the sand the same way each time.
- Measure each crater rim to rim in two directions and take the mean.
- Calculate energy E = mgh for every drop.
- Plot log(diameter) against log(energy). The gradient of the best-fit line is the exponent in D ∝ Eⁿ.
- What to measure
- Mass, height, energy and mean diameter for every drop; the gradient and its uncertainty from the log–log graph.
- How to present it
- A scientific report with your log–log graph, the exponent with an uncertainty, a comparison with the expected value of about 0.25, and an evaluation.
- Take it further
- Does changing the ball's density at the same energy change the crater? Compare your sand craters with the way crater size grows with energy in the site's impact calculator.
Safety
Eye protection on; maximum drop height 1.2 m from a clamped rule; retrieve steel balls immediately; sweep up spills.