From: harper@kauri.vuw.ac.nz (John Harper)
Newsgroups: sci.math
Subject: Re: Help: Trajectory of falling object with drag
Date: 16 Jun 1997 02:18:12 GMT

>Richard Spierenburg (rjas@nlr.nl) wrote:
>: I need help with the following problem:
>: Given a starting condition of an object (height, speed and direction of
>: speed) what is the analytical solution to calculate:
>: 1) time it will take to reach the ground
>: 2) horizontal displacement (range)
>:       drag = K x V2   (proportional to the speed squared)

No exact analytical solution is known unless the trajectory is vertical.
If the initial velocity >> terminal velocity then you can get an approx
analytical solution by assuming the trajectory is in 2 parts:
1. a straight line in which vel >> terminal and you can neglect gravity
2. a vertical line in which the object falls at its terminal velocity.
(In this case the part where the object turns the corner occupies a 
small fraction of the total time and distance.) Remember KV^2 is not an 
exact drag law; merely a fairly good approx. for a certain range of 
speeds depending on body shape. For a sphere it's Reynolds number R
from  1000 to 200,000 or so.
 
To see what actually happens see Encyc.Brit. 1971 ed. "Artillery" (I 
don't have the time to check whether later editions have the same
helpful picture) which shows the trajectories followed by shells from 
various kinds of gun. 

For a practical application of the low-Reynolds-number case (R << 1)
in which drag = k x V see A.H.R. Buller's Researches on Fungi (1909?; 
vols 1-6 Longmans Green; reprinted Hafner NY 1958; vol 7 had different 
publishers but I think this topic is in 1-6 somewhere). He gave the 
(easy) analytical solution because he wanted to understand what happens 
when fungi shoot out their spores. (there are many different kinds of 
spore, but some are spheres and are ejected at low R, and the theory 
applies to them.)

I'm afraid Buller failed to resist the temptation to call the curve 
followed by a low-R spore a sporabola.
 
John Harper School of Math+Comp Sci Victoria Univ Wellington New Zealand