On the popular Mythbusters TV show, one myth to be tested was that in World War II, a gunner fell out of an airplane without a parachute from 22,000 feet, and that he survived because his fall was partially softened by the shock wave from an exploding bomb on the ground just prior to his own impact.

To test the plausibility they dropped a dummy from a height they thought was sufficient to result in the dummy's reaching "terminal velocity", the velocity at which air resistance exactly balances the acceleration of gravity. It's assumed that terminal velocity for a falling human is 120 miles per hour. Co-host Adam did some calculations and determined that terminal velocity would be acquired in 5.5 seconds and require only 500 feet of falling.

What's wrong with the calculation? Assume that the acceleration (call it deceleration if you like) due to air resistance at any given instant is proportional to the square of the downward velocity.

How fast would the falling body be traveling after freely falling 500 feet subject only to gravity and air resistance? How far would a body have to fall to get within 1 mile per hour of terminal velocity? (Again assume terminal velocity to be 120 miles per hour.)

Using an excel spreadsheet and brute force numerical integration, and the following reduced equation, (a time step of 0.001 sec was used).

150/32.2 * dv/dt +.004842*v^2 - 150 = 0 yields the answer of reaching 119 mph after 14.889 sec, and after falling 1958.3 ft.

Not knowing or remembering enough math, I spent some time searching the web for solutions methods to the basic equation of

aY'' + b(y')^2 + c = 0.  This should be related to problems in military ballistics, since cannon shell trajectories would have to be calculated this way.  However, I had no luck.  I would be extremely interested in a closed form solution! On the other hand, I remember a PBS show where during WWII human "calculators" spent days or weeks "summing" to create tables for each specific cannon after a test shot.  Maybe there is no analytical solution?  Also, Eniac, the first digital computer, was used to this purpose (I think).

Edited on December 27, 2006, 4:56 pm

Posted by Kenny M on 2006-12-27 15:40:07