WOBBLrock
How fast does a fist sized rock hit the ground after falling off a 300 m cliff?
OK
Hits the ground 8.72 s after release at 56.8 m/s (205 km/h). Without air it would land after 7.82 s at 76.7 m/s, so drag takes 26% off the impact speed.
fall time
8.72 s
impact speed
56.8 m/s
without air
7.82 s, 76.7 m/s
terminal speed
66.2 m/s
Confidence high
- passed step convergence, hand calculation, vacuum drop, energy balance, units and sanity
Checks
| status | check | result |
|---|---|---|
| pass | step convergence | flown again with a tolerance ten times tighter, the flight time 8.72 s and impact speed 56.8 m/s moved by at most 1e-06% (the impact speed) |
| pass | hand calculation | with the air at its density halfway down (1.207 kg/m3) the terminal speed is 66.2 m/s, and v = vt tanh(g t / vt) with h = vt^2/g ln cosh(g t / vt) gives 8.719 s and 56.9 m/s against 8.717 s and 56.8 m/s from the run (0.16% apart), the difference coming from the air thickening on the way down |
| pass | vacuum drop | with the air taken away and gravity held at its ground value the run lands after 7.822 s at 76.7 m/s, and sqrt(2h/g) gives 7.822 s and 76.7 m/s (1.5e-13% apart). The air adds 0.895 s to the fall and takes 19.9 m/s off the impact |
| pass | energy balance | drag took 1.37 kJ and gravity gave 3.03 kJ, which leaves 1.66 kJ of kinetic energy at impact, and the impact speed carries 1.66 kJ (apart by 4.9e-07% of the largest term) |
| pass | units and sanity | speeds stay under Mach 0.167, 1.03 kg in a 9 cm body, 2700 kg/m3 for a sphere |
Events
| event | time | altitude | downrange | speed |
|---|---|---|---|---|
| release | 0.000 s | 300 m | 0 m | 0 m/s |
| impact | 8.717 s | 0 m | 0 m | 56.8 m/s |
Assumptions
- A point mass in a vertical plane, so it does not spin, tumble or move sideways, and its drag acts against the air rushing past it.
- Drag coefficient 0.6 on a frontal area 9 cm across, the same at every speed.
- The 1976 US standard atmosphere from 0 m above sea level, a standard day.
- Gravity falls off with the square of the distance to the Earth's center, over a flat Earth that does not rotate.
- Still air.