trajectory of a bullet
Jon G.
Excel 97 Worksheet calculates the 2 points:
http://www.geocities.com/jongiff2000/GPS_math.xls
Simply enter the coordinates of the 3 centers and 3 radii, and the Excel
program calculates the coordinates of the 2 points instantly.
Save the Excel file if you'd like and use it for your applications.
For instance, suppose you have a tripod with two fixed legs and one
adjustable leg. The location of the vertex of the tripod can be calculated
from the lengths of the 3 legs and the coordinates of their bases.
===
A gunshot sends a bullet at Mach n sweeping out a cone of half angle c.
Three microphones of known coordinates pick up the shock wave at known time
differences. From this information, what are the coordinates of the gunman?
Find the trajectory of the bullet and predict where it will go before it
gets there.
This problem has been solved and its solution is known and used on
helicopters by the military. Can you solve it? I've tried and I can't.
Find the vector equation, differentiate it with respect to time, and go from
there.
A 4th microphone is used to determine if the gunshot originated from above
or below the plane of the first 3.
Matt
A bullet travels in approximately a straight line. How can it "sweep
out a cone"? Don't understand the question.
Jasen Betts
>> A gunshot sends a bullet at Mach n sweeping out a cone of half angle c.
>
> A bullet travels in approximately a straight line. How can it "sweep
> out a cone"? Don't understand the question.
I think the inquirer means that the bullet produces a cone-shaped
shock-wave that has the bullet at its apex.
http://en.wikipedia.org/wiki/Sonic_boom
http://en.wikipedia.org/wiki/Suppressor#Suppressors_vs._Silencers
>> Three microphones of known coordinates pick up the shock wave at known time
>> differences. From this information, what are the coordinates of the gunman?
>> Find the trajectory of the bullet and predict where it will go before it
>> gets there.
the input has three degrees of freedom (t₁,t₂,t₃)
the shooter had 6 (t₀,x,y,z,θ,Φ)
there's not sufficient information.
>> This problem has been solved and its solution is known and used on
>> helicopters by the military.
no, if that's so, some other problem has been solved.
>> Can you solve it?
noone can, it's iompossible.
>> A 4th microphone is used to determine if the gunshot originated from above
>> or below the plane of the first 3.
still not enough, six microphones would be needed
bye.
Robert Israel
> On 2008-12-07, Matt <matt271...@yahoo.co.uk> wrote:
> >> A gunshot sends a bullet at Mach n sweeping out a cone of half angle c.
> >
> > A bullet travels in approximately a straight line. How can it "sweep
> > out a cone"? Don't understand the question.
>
> I think the inquirer means that the bullet produces a cone-shaped
> shock-wave that has the bullet at its apex.
>
> http://en.wikipedia.org/wiki/Sonic_boom
> http://en.wikipedia.org/wiki/Suppressor#Suppressors_vs._Silencers
>
> >> Three microphones of known coordinates pick up the shock wave at known
> >> time
> >> differences. From this information, what are the coordinates of the
> >> gunman?
> >> Find the trajectory of the bullet and predict where it will go before it
> >> gets there.
>
> the input has three degrees of freedom (t₁,t₂,t₃)
> the shooter had 6 (t₀,x,y,z,θ,Φ)
I can't make sense of your non-Ascii characters, but I assume
the inputs are times t_1, t_2, t_3, the shooter's six degrees of
freedom are three position coordinates, two direction coordinates
and the time of shooting.
> there's not sufficient information.
>
> >> This problem has been solved and its solution is known and used on
> >> helicopters by the military.
>
> no, if that's so, some other problem has been solved.
>
> >> Can you solve it?
>
> noone can, it's iompossible.
>
> >> A 4th microphone is used to determine if the gunshot originated from
> >> above
> >> or below the plane of the first 3.
>
> still not enough, six microphones would be needed
Even six microphones wouldn't distinguish between different starting positions
on the same trajectory.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
David Bernier
> On 2008-12-07, Matt <matt271...@yahoo.co.uk> wrote:
>>> A gunshot sends a bullet at Mach n sweeping out a cone of half angle c.
>> A bullet travels in approximately a straight line. How can it "sweep
>> out a cone"? Don't understand the question.
>
> I think the inquirer means that the bullet produces a cone-shaped
> shock-wave that has the bullet at its apex.
>
> http://en.wikipedia.org/wiki/Sonic_boom
> http://en.wikipedia.org/wiki/Suppressor#Suppressors_vs._Silencers
>
>>> Three microphones of known coordinates pick up the shock wave at known time
>>> differences. From this information, what are the coordinates of the gunman?
>>> Find the trajectory of the bullet and predict where it will go before it
>>> gets there.
>
> the input has three degrees of freedom (t₁,t₂,t₃)
Depending on non-trivial character encoding characteristics of
news-readers, Greek letters, subscripted quantities and so on
can be represented as gibberish.
The 3 times can be written as:
(t_1, t_2, t_3) .
> the shooter had 6 (t₀,x,y,z,θ,Φ)
or:
the shooter had 6 (t_theta, x, y, z, theta, phi).
David Bernier
Matt
> On 2008-12-07, Matt <matt271829-n...@yahoo.co.uk> wrote:
>
> >> A gunshot sends a bullet at Mach n sweeping out a cone of half angle c.
>
> > A bullet travels in approximately a straight line. How can it "sweep
> > out a cone"? Don't understand the question.
>
> I think the inquirer means that the bullet produces a cone-shaped
> shock-wave that has the bullet at its apex.
Yeah sorry inquirer, being a bit slow...
>
> http://en.wikipedia.org/wiki/Sonic_boom
http://en.wikipedia.org/wiki/Suppressor#Suppressors_vs._Silencers
>
> >> Three microphones of known coordinates pick up the shock wave at known time
> >> differences. From this information, what are the coordinates of the gunman?
> >> Find the trajectory of the bullet and predict where it will go before it
> >> gets there.
>
> the input has three degrees of freedom (t₁,t₂,t₃)
> the shooter had 6 (t₀,x,y,z,θ,Φ)
>
> there's not sufficient information.
>
> >> This problem has been solved and its solution is known and used on
> >> helicopters by the military.
>
> no, if that's so, some other problem has been solved.
Could it be that the system just determines the direction the bullet
came from, assuming it's directed somewhere towards the microphones?
(Without slogging through it I'm not sure how sensitive that
"somewhere towards" would be though.)
Jasen Betts
> Could it be that the system just determines the direction the bullet
> came from, assuming it's directed somewhere towards the microphones?
> (Without slogging through it I'm not sure how sensitive that
> "somewhere towards" would be though.)
given that any three (non co-linear) points define a circle
shots arriving from the same origin but aimed slightly differently
will give different shockwave arrival times.
arrival times of the shock wave tell a lot more about where the shot
was going than about where it came from.
If the microphones are improved to give directional and/or distance
information as well as time information that increases the degrees
of freedom in the input tuple which would help a lot
as someone else has pointed out there are only two degrees of freedom
for the location of the shooter (I had three) as it could lie anwhere along the
previous path of the bullet, however there is another degree of
freedom for the shot, it's speed, so we still need atleast 6 degrees
of freedom in our input (or some magic geometry that cancels out the
speed and direction the bullet is going in) to hope to resolve the
shot to it's origin.
David Bernier
At the end of the original post, it says:
"A 4th microphone is used to determine if
the gunshot originated from above or below
the plane of the first 3."
Also, maybe the Mach number is given as n. Since they mention cones,
I think we can forget about gravity. If a Concorde
flies straight at Mach 2, I think the shock-wave looks like
a cone advancing in the same direction as the plane.
The Mach number would determine how pointed the cone
is. Actually, maybe we can think of a Concorde flying
in a straight line , not necessarily horizontal.
David Bernier