### Re: how far can a bullet travel shot into water?

Date: Thu Jul 12 13:06:51 2001
Posted By: Jaime Valencia-Rodríguez, Guest Researcher, Chemical Science and Technology Lab, NIST.
Area of science: Physics
ID: 991288529.Ph
Message:
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Dear David:

The problem you are addressing is an old one and it looks deceivingly
simple. We need to start mentioning Newton's Second Law. In the following
ballistic trajectories

computing and visualizing ballistic trajectories. A ballistic trajectory
is the path followed by an object which, after it is given some initial
velocity, travels only under the influence of gravity. For the purposes of
this lesson we will ignore the effects of air resistance."

Well, ignoring air resistance, or, more generally, the medium resistance,
is a huge simplification. Nevertheless, we will go along for the moment.

The Ballistic Trajectory Problem
One of the first problems studied in an introductory physics class is the
ballistic trajectory problem. Let's assume that we are standing in a flat
field and that we throw a ball so that it starts out moving at some
velocity V > 0 and at some angle theta radians with the ground. At any
given time t, the ball's horizontal distance is given by

Vt cos (theta)

and its vertical height is given by

Vt sin(theta) - (1/2)gt^2

where g is the earth's gravitational constant, or 9.8 meters/sec/sec. This
assumes that the ball's initial horizontal and vertical positions are both
0. It will prove convenient to define functions for distance and height
that take initial velocity, initial angle, and time as arguments and
return the distance and height of the ball.

distance := (V, theta, t)     ->      V*t*cos(theta)
height := (V, theta, t)     ->     V*t*sin(theta) - .5*9.8*t^2;

So far, so good. It is clear that if we shot a weapon horizontally (i.e.,
theta = 0), after certain time, the bullet will hit the floor. This is the
so called range of the weapon.

A nice link presenting interactive graphics is the following:
http://www.mcasco.com/p1anlm.html

A more formal approach could be find in:
http://www.phys.uidaho.edu/~pbickers/Course
s/310/Notes/part2/node20.html

Now, since in real life we could not ignore the drag due to the air (or
water) resistance, we need to refine our ideas.
It is evident that the drag is a force opposing the movement of
the bullet. How big is this force?. In
http:
//www.i3solutions.com/vehiclesimulation/dragequations.htm

we find:

Air (water) Friction:

Fair = 1/2 Cd A đ V2

Rolling Resistance:

Rolling = (RRConst + Velocity * RRXcoef) * Weight * Cos(slope)/100

Total Losses:

FTotal = Fair + FRolling + FGrade

where Cd is coefficient of drag, A is the cross sectional area, V is the
velocity, đ is the density of the medium.
It is evident that the bigger the density the shortest distance traveled
by the bullet.

Finally, in
http://www.las
cruces.com/~jbm/ballistics/secdens.html

we find:

Sectional Density and Ballistic Coefficient
The BC, or ballistic coefficient is defined as:
BC = w / [i d2]
where the diameter is specified in inches and the weight in pounds and the
form factor is found using:
i = CD / CDG
The sectional density is defined as:
SD = w / d2
making the ballistic coefficient
BC = SD / i
So this means that the ballistic coefficient is proportional to the weight
of the bullet and inversely proportional to the diameter squared. (Keep in
mind that the ballistic coefficient is also inversely proportional to the
form factor which depends on the shape of the bullet!)
Calculation of the sectional density is straight forward. For a 300
grain, .338 caliber bullet, the sectional density is:

SD = [ 300 gr / (7000 gr/lb) ] / [ 0.338 in ]2 = 0.375 lb/in2

NOTE: With the common definition of the sectional density, the units have
to be converted when used with drag functions, velocity, etc, to convert
the in2 to ft2 resulting in a factor of 144.

Variables
d 	bullet diameter 	w 	bullet weight
SD 	sectional density 	BC 	ballistic coefficient
i 	form factor 	G 	"G" function
CD 	drag coefficient 	CDG 	drag coefficient of the standard
bullet

The numerical answer, of course, depends of the particular values of all
this parameters.

There is a nice book devoted to such problems. It could be found in:
http://www.border-
barrels.com/book.htm

I hope this helps

Regards

Jaime Valencia

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