I?m designing a postcard to send out later this year that references a
famous photograph by Dr. Harold Edgerton of a bullet shooting through
an apple.

Half the image will be physics notes on graph paper, and the other
half will be a photorealisitic image of the event.

The problem is, my target audience knows physics but I don?t, so I can?t fake it.

I need to make some real notes, measurements and equations that I can
put on the graph paper side of the image.

My question is, ?How, specifically, should I mark up this image to
make it look as if it were some kind of physics assignment.

Please see my sketches here:

http://temporary.printfreegraphpaper.com/1.jpg
http://temporary.printfreegraphpaper.com/2.jpg

and my blank graph paper here:

http://temporary.printfreegraphpaper.com/3.jpg

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Request for Question Clarification by pafalafa-ga on 05 May 2005 19:03 PDT

You could try this (unless it absolutely has to be apple-related):


http://xtrsystems.com/strobe/strobe_trigger

Hi,
Do the notes have to be physics related, or just mathematical/geometrical?
Either way, you would probably like them to relate to the semicircle
completing the apple.
If the notes could be geometrical, that would make things a lot easier:
A radius line on the semicircle and a "cord", and then a couple of
circle-related formulae:  area, circumferance, smaller area defined by
the cord as a function of the angle of the radii to its ends.  (Have
to find them yourself. :-( )

If the notes do have to be physics related, someone else will have to help you.
(Where's mathtalk-ga ?)

NIce idea for the card.

Thanks for your comment?s myoarin, but I think I want to keep the card
physics based.  While my target is a math audience so really any
science or math could work, I want to stress the bullet, and physics
seems to address this motion the best.  Something like the trajectory
path of the bullet through the apple.  Also, I?d like something that
uses the graph paper, not some problem that just happens to be done on
graph paper.

I just don?t think it would be as strong if it were an apple with the
circumference measurements on one side with the area figured out, and
then for the real life picture to have a bullet coming out of the
apple.  I think the point would be lost or really confusing.

Again, I would like to stress the trajectory paty of the bullet on the
graph paper side.

Yes, a very attractive concept, somehow reminiscent of Rene Magritte.

I'm thinking maybe something along the lines of conservation of
momentum laws?  It would suitable vector calculus symbolism.

regards, mathtalk-ga

[Conservation of Momentum]
http://www.grc.nasa.gov/WWW/K-12/airplane/conmo.html

Note that this is a very simple (but nonlinear) equation, Euler's eqn. in 1-D:

    dp       du
  - -- = r u --
    dx       dx

The "more complex" link given there references the Navier-Stokes
equations for unsteady flow, which may be more aesthetically pleasing:

[Solving the unsteady Navier-Stokes Equation by Gunnar A. Staff]
http://heim.ifi.uio.no/~kent-and/inf5670/lecture_notes11.pdf

From the opening paragraph of Section 1:

  ?u                1
  -- + u · grad u + - grad p = ? ?u + f
  ?t                ?

which is paired with the "divergence free" constraint (incompressibility):

  div u = 0

Here u is a vector quantity (the flow velocity at each point in the
domain) while pressure p is a scalar quantity, so in a hand-drawn
equation I'd probably put a little "arrow" (pointing right) over the
top of each u.

Now Navier-Stokes governs fluid flow.  The "rational mechanics" of an
apple are of course not quite so "simple", since it has an elastic
skin in addition to a pulpy interior.  If you like I could give this
some further thought.


regards, mathtalk-ga

Hmmm!?  I feel a little responsible for calling in the Big Gun on this.
And then like a Magritte-William Tell - right on the apple.
Mathtalk, I like to think that you can just pull formulae like that
out of your sleeve, but it still is an effort to post them as a
comment.
I would pay you if I could in appreciation for your responding.  Hint, Hint.  :-)

Thank you guys, you?re full of help.

Mathtalk, I?m not a physics person, so a lot of those things aren?t
registering with me.  But I pulled your formula and a graph from the
link.  You said something about putting right arrows above the u?s but
I didn?t  know where to put the u?s.  Are you talking about the u?s in
the formula?  Again, I have no idea what I?m doing, but I made up
another sketch here:

http://temporary.printfreegraphpaper.com/4.jpg

Myoarin, I agree about how helpful Mathtalk is, in fact you?ve both
been helpful.  But I don?t understand the hints.  I thought one can
only pay and tip another person if they actually place an answer post,
not a comment post.  Am I wrong?

graphpaper2-ga,
You are right, but you can post a comment or clarification asking a
researcher to post his work as an answer.  See FAQs.  And since you
seem to be wanting to use mathtalk's info, and asking for a little
more detail, this would seem appropriate.

(Between you and me:  I don't understand the formulae either :-)

On Myoarin?s suggestion and if you?re interested, I invite you
MathTalk to provide a revision to your posted comment as an answer
that tells me ?how, specifically, should I mark up this image to make
it look as if it were some kind of physics assignment.?

If you find it easier than explaining and you would like to, you can
legibly markup a copy of my blank graph image and submit a link to it.
 I would consider that very sufficient.
 
http://temporary.printfreegraphpaper.com/3.jpg

If you're looking for more equations to fill space that are somewhat
related, you can use m1iv1i + m2iv2i = m1fv1f + m2fv2f where m is the
mass and v is the velocity, 1 is for the first object, i is for
initial(pre-collision), f is for final (post-collision). That would
look something like

(0.0015kg * 600m/s) + (0.1kg * 0m/s) = (0.0015kg * 500m/s) + ( 0.1kg + 0.15m/s)

Just an example as I doubt the bullet would slow that much, and the
apple would obviously lose mass, each part having its own velocity
(which wouldn't always be in the x direction), but it gives you the
general idea.

Weights (along with conversions) and speeds pulled from
http://www.winchester.com/products/catalog/components/riflebullets.aspx
and 
http://www.recguns.com/Sources/VIIE8.html
http://dictionary.reference.com/search?q=measurement
http://hypertextbook.com/facts/2004/WaiWingLeung.shtml

Just as an idea, put in specs for everything you could know about the
picture - bullet speed, weight, dimensions, speed of rotation; apple
weight, skin meat and core thickness/density.

Also, something most physics-related people would appreciate, create a
free body diagram.  http://laser.phys.ualberta.ca/~freeman/enph131/fbd.htm
Understandably, if you are not into physics it wouldn't be the easiest
thing to create, but just the tracings of one could suffice.

As far as the trajectory of the bullet's path, I can't really see much
that you could do.  Plot a coordinate plane, draw an arrow in the
direction it's traveling, write the speed, show that the bullet is
rotating with given speed.  If you really want to get into it, you
could supply the diameter of the apple, find out the time it takes for
the bullet to travel that distance (which would be something insanely
short) and plug it into the basic kinematic formula of s = si + vi*t +
1/2 a*t^2, which s is distance, si is inital position, vi is inital
velocity, t is the time it takes to travel that distance, and a is the
acceleration on the object (-9.81 m/s^2 (the force of gravity)).

/trying to be helpful

Thank you very much racic-ga for your help.

I invite you to post your comment as the answer.

I took your advice and created this image:
http://temporary.printfreegraphpaper.com/5.jpg

In your posting of your comment as the answer, I would greatly
appreciate any feedback you have on the image.

Thanks again.

Wow, just noticed that I screwed that equation up, I apologize.

This: (0.0015kg * 600m/s) + (0.1kg * 0m/s) = (0.0015kg * 500m/s) + (
0.1kg + 0.15m/s)

should be: 
(0.0015kg * 600m/s) + (0.1kg * 0m/s) = (0.0015kg * 500m/s) + ( 0.1kg * 0.15m/s)

You can also put things such as:
J hat = delta p = m delta v = 0.0015kg(500m/s - 600m/s) = -0.015J.

By hat I mean a little arrow over the top of the letter, which denotes a vector.

Your call on this one, but I'd say you should put the math down in a
more linear manner, as if it were actually worked, but that would take
up less space.  Put boxes around any possible answers, for style as
much as clarity.  The timesteps are a nice touch.