A simple scaling experiment - just something to think about

[dongfang]

Hi,

Imagine a spud gun with

barrel length l
chamber pressure p
projectile mass m
barrel cross section area A

Disregarding this and that and a lot of other things, and assuming constant pressure and acceleration through the barrel, we get:

let t be the time from firing till the projectile leaves the muzzle.

l = 1/2 a t² ==> t = sqrt(2l / a)
(a = acceleration)

The force on the projectile:

F = ma = pA

so a = pA / m.

and the muzzle velocity is:

v = at = a * sqrt(2) * sqrt(l) / sqrt(a) = sqrt(a) * sqrt(2) * sqrt(l) =
sqrt(p) * sqrt(A) * sqrt(2) * sqrt(l) / sqrt(m)

Now imagine scaling up the gun by a factor r:
l' = r * l
m'= r³ * l³ (we scale the projectile by l in each dimension; it becomes cube l times heavier)
A' = r² * A
p' = p (same pressure)

The muzzle velocity of the scaled gun becomes

v'= sqrt(p') * sqrt(A') * sqrt(2) * sqrt(l') / sqrt(m')
=sqrt(p) * sqrt(A) * sqrt(r²) * sqrt(2) * sqrt(l) * sqrt(r) / (sqrt(m) * sqrt(r³))
= v

So, disregarding this and that and a lot of other things, and assuming constant pressure and acceleration through the barrel, we get:

- there is no muzzle velocity gain in scaling up a gun.

That surprises me a little (or does somebody see a miscalculation somewhere?)

Maybe it's really so; don't pistols shoot about as fast as howitzers?

Rgds
Soren

[jrrdw]

Hi,

Imagine a spud gun with

barrel length l
chamber pressure p
projectile mass m
barrel cross section area A

Disregarding this and that and a lot of other things, and assuming constant pressure and acceleration through the barrel, we get:

let t be the time from firing till the projectile leaves the muzzle.

l = 1/2 a t² ==> t = sqrt(2l / a)
(a = acceleration)

The force on the projectile:

F = ma = pA

so a = pA / m.

and the muzzle velocity is:

v = at = a * sqrt(2) * sqrt(l) / sqrt(a) = sqrt(a) * sqrt(2) * sqrt(l) =
sqrt(p) * sqrt(A) * sqrt(2) * sqrt(l) / sqrt(m)

Now imagine scaling up the gun by a factor r:
l' = r * l
m'= r³ * l³ (we scale the projectile by l in each dimension; it becomes cube l times heavier)
A' = r² * A
p' = p (same pressure)

The muzzle velocity of the scaled gun becomes

v'= sqrt(p') * sqrt(A') * sqrt(2) * sqrt(l') / sqrt(m')
=sqrt(p) * sqrt(A) * sqrt(r²) * sqrt(2) * sqrt(l) * sqrt(r) / (sqrt(m) * sqrt(r³))
= v

So, disregarding this and that and a lot of other things, and assuming constant pressure and acceleration through the barrel, we get:

- there is no muzzle velocity gain in scaling up a gun.

That surprises me a little (or does somebody see a miscalculation somewhere?)

Maybe it's really so; don't pistols shoot about as fast as howitzers?

Rgds
Soren

WTF...Give me 1+1=2. As smart as you just made yourself look with these figures, can you translate it to simple math? For real! Can you?

[Hotwired]

Cannons can have much greater muzzle speed because they can take more recoil.

I think they can chuck stuff at about mach 5, I'll check though. Rifles and pistols don't go much past the sound barrier though.

With scaling up everything I'd not expect to see much change in speed but its uncommon for anyone to scale up a cannon like that. Different sizes tend to have different optimal layouts.

[us sniper]

The muzzle velocity shouldn't increase with the bigger gun. The projectiles fired is also proportional to the gun. The cannons fire futher because of the inertia behind their rounds.

[boilingleadbath]

Jrrdw, that is basicaly as simple as the math is going to get... if you get scared of a square root sign or something raised to the third power, you should just stay away from that type of discussion.

That said:

1) Howitzers are designed to shoot somewhat slowly, because they arn't line-of-sight weapons... but, yeah, their velocity is about that of a (typical) handgun. (I'm not counting .223 "pistols" with 12" barrels, for instance)

2) The math looks right, and provides the answear I'd expect.

3) Here's a slightly different derivation of the answear:
a) E = 1/2MV² (kinetic energy)
b) E = FD = R²D (applied energy is force (radius squared) times distance)

and we assume they are equal and ignore that 1/2 (it's a useful constant, but we don't care about getting the real velocity.)
MV²=R²D
V = sqrt(R²D/M)

And, hey, let's expand everything by r!
V = sqrt(R²r²Dr/Mr³)
The rs cancle out

4) But, what the hell makes us think that if the cannon is twice as long, we'll be putting ammo twice as long in it?
If we don't, of course, the rs don't cancel out compleatly, and V ends up being dependent appon sqrt(r).

5) interesting point:
in this analysis...
R²D = Volume
So, that's sqrt(volume/mass)

...hot dang, that's EVBEC!

[dongfang]

Hi boilingleadbath

E = R² D ? Never heard abt this before. Or I forgot something. Are you referring to a law of physics that I should know ?

And, EVBEC, what is that?

Rgds
Soren

[dongfang]

Ahh got it; you just say that F and R² are proportional. Pressure is part of their ratio.

[jrrdw]

Jrrdw, that is basicaly as simple as the math is going to get... if you get scared of a square root sign or something raised to the third power, you should just stay away from that type of discussion.

Not much in this life scare's me anymore, i've grown to that point. I asked for a translation to simple math, maby i ask to much?

[boilingleadbath]

EVBEC is a spreadsheet I wrote... and it basicaly uses that equation to extrapolate from the latke data.
Search the forums and our wiki for it and you'll find a couple thosand words on the topic.

No, Jrrdw, you didn't ask too much - you asked the <i>impossible</i>.

[jrrdw]

I just emailed it to my niece in collage, she should be able to convert it into simpleton math for us uneducated folks. I'll get back to ya's with the results one way or the other, if anybody can do it, she can! Thiers got to be away!!!