Deriving the maximum range and angle of a projectile

In a thread in the Green group, AnarchistAsian and I were discussing his coil guns. I posed the question of what range he could get, and he asked me to go through the physics derivation. Here it is, simplified to require just algebra and trig.

The kinetic energy of the projectile E = ½mv2, gives v = sqrt(2E/m) as the speed at launch. Let θ be the angle at which you launch (θ=0° is horizontal, θ=90° is straight up). Then you can decompose the speed into two components:

vh = v cos(θ) is the horizontal speed,
vv = v sin(θ) is the vertical speed.

Gravity only pulls vertically, so the projectile's vertical speed will be slowed down until it reaches its maximum altitude, then it will fall back until it hits the ground. The horizontal speed will remain constant until it hits the ground and stops. To figure out the range, you need to know the time t that the projectile flies before it hits the ground; then the range is just

R = vht

Energy conservation guarantees that it's downward speed at the end is equal to its original upward speed, just with a change of sign. That also means that the total flight time of the projectile will be half going up and half going down. Once you determine how long it takes to reach the top of its flight, you're done; just double that answer :-)

In the vertical direction, the maximum height

H = vvt - ½gt2

(you need calculus to derive this result). From energy conservation, the initial kinetic energy in the vertical direction, Ev = ½mvv2 must equal the potential energy at the top of the flight, Pv = mgH:

mgH = ½mvv2
H = ½vv2/g

Substitute this on the left hand side of the trajectory expression,

½vv2/g = vvt - ½gt2
vv2 = 2gvvt - g2t2
vv2 - 2gvvt + g2t2 = 0
(vv - gt)2 = 0
vv = gt

So, t = vv/g = v sin(θ)/g is the time to reach maximum height. Double that as discussed above, and you get the range R = 2 sin(θ) cos(θ) v/g.

Work out the angle that gets you maximum range by just plugging in different angle values and finding the one that is biggest.

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Goodhart9 years ago
Not meaning to interupt nor disrupt, but I am having a little difficulting understanding the following portion of the equation: (Y=0° is horizontal, Y=90° is straight up).
kelseymh (author)  Goodhart9 years ago
Ignore the italic "angstrom" symbol :-( I'bles WikiFormatting processor has some stupid problems with the extended ASCII characters (character codes 128 to 255). It sticks in extra symbols, and then if you try to edit the text to get rid of them, it actually adds even more of them!

Those two statements are supposed to be "Y = 0 degrees is horizontal, Y = 90 degrees is straight up", where I'm using "Y" as the symbol for angle instead of Greek "theta".
PKM kelseymh8 years ago
The hacky workaround for the degree symbol is a superscript o, put chevrons on either side. So
Θ / θ . hmmmmm works for me. ;) . #920 and #952 . . Found that by Googling "HTML +theta. Seems I have a lot more stuff to add to my formatting topic. :)
kelseymh (author)  NachoMahma9 years ago
Thanks. Those are standard Unicode values? You should be wary before going down the road of adding all of them -- there are 65,536 possible! Then there's Big 5 and the other asian fonts...
> Those are standard Unicode values?
. I dunno. Found 'em at Entities for Symbols and Greek Letters. That page says "Glyphs of the characters are available at the Unicode Consortium," so I'm guessing they are.
> ... there are 65,536 possible!
. What do you think about having a spreadsheet for download? Using Fill and C&P, it shouldn't take too long to build - that's what I did for the ASCII chart screen captures.
. I'm guessing that a LOT of the characters are not needed (eg, any pictograph language chars). For use on Ibles, just the English (Roman?), Greek, and Math chars should be enough. Probably ought to include any German, French, &c chars for those times one wants to spell Gotterdammerung or tete-a-tete properly.
kelseymh (author)  NachoMahma9 years ago
German, French and Spanish are already part of the ISO Latin-1 encoding (the table you have :-). They all have proper entity names, as well as numeric values: é = é ö = ö and so on.

It's the non-Latin characters for which you have to go to Unicode (Greek, Cyrillic, the Cyrillic-derived Eastern Europeans, etc.).
. Does that mean that all I need is a supplement to the ASCII/Latin-1 chart, with the Greek Unicode characters? Or do you think there is more that should be added? . I don't want to be Anglo-centric, but the charts may be too big if the "minor" (eg, Cyrillic) languages are included. With a SS, it wouldn't be a problem. Arrrrgggghhhh! We need tables in the markup! Then the user could copy the entity directly from the page. . heehee Yeah, Latin. Not sure where I got Roman from.
kelseymh (author)  NachoMahma9 years ago
I wouldn't do it as a supplement -- that's an open ended problem. I would make mention of the extended characters, and provide a link to a definitive site (as you provided above). The alphabet is "Roman", the character set is ISO Latin-1 :-)
kelseymh (author)  NachoMahma9 years ago
Could you add that link to your FAQ? Thanks!
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