Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: drag coef. and falling bullets



Regarding John D.'s advice to Leon:

... You
might be much better off dropping bullets from a tall building. A
multi-story indoor atrium or stairwell alleviates wind problems. Let them
hit paper targets and see if you get point-first or broadside holes.

and Leon's response:

We're off to find a stairwell. Thanks for the
information!

It seems to me that Leon is unlikely to find a stairwell that it deep
enough to properly determine the terminal velocity of his bullets unless
his velocity measurement apparatus is capable of accurately measuring the
residual acceleration of the bullet because it is quite likely that the
bullets will not have achieved a decent fraction of the terminal velocity
by the time they reach to bottom of the well.

If we assume that the drag on a falling body depends quadratically on the
body's speed we can exactly solve for the distance, z, fallen (from rest)
in order to achieve a speed v given a terminal speed of V. The resulting
formula is:

z = - (V^2/g)*(ln(1 - (v/V)^2))/2

We see that the relevant characteristic distance, z_0 == V^2/g , sets the
scale for the distance z needed to achieve a speed ratio of v/V. For
instance, the table below matches various values of z/z_0 with the
corresponding values of v/V.

v/V z/z_0
0.50000 0.14384
0.80000 0.51083
0.90000 0.83037
0.92987 1.00000
0.95000 1.16395
0.99000 1.95852
0.99500 2.30384
0.99900 3.10755
0.99950 3.45400
0.99990 4.25862
0.99995 4.60518
0.99999 5.40989

We see that we need an available distance of z = z_0 == V^2/g in order
to have the bullet get up to just 93% of its terminal speed. Since we
expect the terminal speed to be around a few tens of meters per second
this means we may need a very deep stairwell or a very tall building from
whose roof the bullets are to be dropped. For instance, if we use Brian
W.'s estimate of V = 35 m/s this gives z_0 = 125 m (~40 stories). If we
had a terminal speed of 54 m/s (comparable to a prone human body falling
at terminal speed) we have z_0 = 298 m which corresponds to the height of
a tall skyscraper.

If we use the info reported by John D. from Wegner's book:

Using figure 7.3 on page 89 of Wegener, we have
0.47 -- sphere -- representative of point-first flight
1.17 -- cylinder -- representative of broadside flight

and take the 1.17 value for the drag coefficient and use the weight and
dimensions for the bullet reported by Brian W. (i.e. 0.073N and
9 mm x 27 mm) and take the air density to be 1.29 kg/m^3 we get a value
of V = 20 m/s. This gives a value of z_0 = 41 m (13-14 stories). If
Leon wants v/V to be within 1% of unity he will need to use a stairwell
whose depth is at least z = 2*z_0 = 82 m (~27 stories).

I hope Leon has access to the necessarily deep stairwell or can
accurately measure the residual acceleration the bullets will still have
when they reach the bottom.

My alternate suggestion for using a stairwell (the deeper the better) as
a laboratory is to monitor the bullets as they fall past multiple
vertically spaced locations (using photogates for instance) and use the
timing information to find a terminal speed/drag coefficient that
reasonably well fits the data for the measured trajectory which, as
function of time, ought to obey the equation:

z(t) = (V^2/g)*ln(cosh(g*t/V))

if the drag really is quadratic in the speed. So, if we define
x = ln(g*t) - ln(V) and define y = ln(g*z) - 2*ln(V) these variables
will obey the universal equation:

y = ln(ln(cosh(exp(x))).

The effect of fitting the data to the universal curve involves
horizontally translating x by some fixed amount and vertically
translating y by twice that amount until the data line up with the
universal curve. The needed translation to make the data fit is just the
logarithm of the terminal speed. If no translation will fit the curve
to the data then either the experimental procedure was compromised by
experimental errors, or the bullets do not fall with a drag proportional
to the square of their speed through the air. Something useful or
interesting is bound to be discovered.

Of course, Scott G.'s method might be a lot simpler to instrument if one
has the high speed fan, venturi air speed detector, and other equipment.

David Bowman
David_Bowman@georgetowncollege.edu