Hands-On Math

Last Revision March, 2011

Last Revision March, 2011

He employed a vertical wind tunnel and adjusted flow rate until drops were suspended to determine terminal velocities. He found that fall speed increased with drop radius up to a radius of about 2.25 after which it increased more slowly and did not increase beyond 8 metres per second. He deduced that the changes in shape increased the air drag thus slowing the fall rate.

For mention of Lenard's raindrop results, see here.

That author uses the term

Stones in the sample ranged in radius from 0.00315 to 0.0077 metres.

4/3 * 3.14159265 * 10

Its mass is:

998.2 * 4.189 * 10

Take the drag Cd of a sphere as ~ 0.4

When these values are plugged into the calculator it provides a vt of ~7.376 metres/s. As steady state fall is our interest, we use this as the chosen velocity to get R as ~ 492. This Reynolds number is marginally low for the Rayleigh approximation to serve and much too high for Stoke's law to apply.

Then the volume is ~4.771 * 10

Flattening was said to begin when the radius exceeded 1.0 mm. The Aerodynamic Database suggests a drag coefficient of 1.42 for extreme flattening, a hollow semi sphere. Choose a drag of 0.85.

The calculator returns a vt of ~ 7.59 metres/s. This leads to an R of ~ 1138 which is nicely within the Rayleigh range.

The calculator returns a vt of ~ 7.736 metres/s. This leads to an R of ~ 1418.

The three estimations foregoing are reasonably in agreement with the observations of Philipp Lenard made over a century ago.

Although one is quite sure that the Rayleigh approximation will not apply to 10-micron fog droplets, curiosity drives us to view the calculator's results for that case.

The volume is ~ 4.189 * 10

The calculator returns a vt of ~ .738 metres/s leading to an R of ~ 0.49. R is much too small for the Rayleigh approximation to apply and the calculated vt is much larger than the observed value of ~ 0.01 metres per second.

With the reported

The agreement of our calculator vt with the reported value of ~ 8.5 metres/s is quite close.

The largest hailstone had a radius of .0077 metres. Its volume and mass are ~ 1.912 * 10

For this case the calculator provides a vt of ~ 10.272 m/s and an R of ~ 5273. The terminal velocity converts to ~ 37 km/hr. Larger stones have occurred and in some case caused considerable property damage.

An air filled volleyball has a mass of about 0.0275 and a radius of about 0.108 metres. With a Cd of 0.4,according to the calculator, the ball would drop 50 metres in ~ 9.4 seconds.

Now put the ball on a scale and inject enough water to double the mass and presume that it is dropped again. In this case the calculator reports that the ball would fall 50 metres in ~ 6.9 seconds.

How long would it take for an iron ball to fall the same distance? Answer: ~ 3.2 seconds. Check it out!

Note that buoyancy is not inherently taken into account by this calculator. The second topic of this chapter, under

Click

Top | Prev. Topic | Topics | Next Chapter |