## 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 = ½mv

v

v

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 = v

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 = v

(you need calculus to derive this result). From energy conservation, the initial kinetic energy in the vertical direction, E

mgH = ½mv

H = ½v

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

½v

v

v

(v

v

So, t = v

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

The kinetic energy of the projectile E = ½mv

^{2}, 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:v

_{h}= v cos(θ) is the horizontal speed,v

_{v}= 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 = v

_{h}tEnergy 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 = v

_{v}t - ½gt^{2}(you need calculus to derive this result). From energy conservation, the initial kinetic energy in the vertical direction, E

_{v}= ½mv_{v}^{2}must equal the potential energy at the top of the flight, P_{v}= mgH:mgH = ½mv

_{v}^{2}H = ½v

_{v}^{2}/gSubstitute this on the left hand side of the trajectory expression,

½v

_{v}^{2}/g = v_{v}t - ½gt^{2}v

_{v}^{2}= 2gv_{v}t - g^{2}t^{2}v

_{v}^{2}- 2gv_{v}t + g^{2}t^{2}= 0(v

_{v}- gt)^{2}= 0v

_{v}= gtSo, t = v

_{v}/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.

active| newest | oldestÂ° is horizontal, Y=90Â° is straight up).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".

becomes

C

^{o}. 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.

It's the non-Latin characters for which you have to go to Unicode (Greek, Cyrillic, the Cyrillic-derived Eastern Europeans, etc.).