# Terminal Velocity of Projectile

## Main Question or Discussion Point

Recently on an airsoft forum I took part in a discussion about the muzzle energy of airsoft guns, which turned into a discussion about the terminal velocity of a .25g BB. Now, I believe that if the muzzle energy of the gun permits, a BB can be launched at 1870fps. Afterwhich, the air acts as a force oposite the direction of motion, thus slowing the projectile to it's terminal velocity, where the air resistance is equal to the gravitational pull on the object.

The other guy thinks it's impossible, and that the BB would turn into a poof of dust...

Am I missing something? Thanks in advance for any help. :-)

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Maybe

You're going to have to be a bit more specific. Let's neglect the initial velocity of the BB for a moment, since it's irrelevant for the discussion of terminal velocity. Assume you're dropping it from a very tall building:

$$md^2x/dt^2 = -gm + m\rho dx/dt$$

Friction is modeled as a force which is proportional to the velocity of the particle in these cases. The terminal velocity will occur where
$$d^2x/dt^2 = 0 = -g + \rho dx/dt \rightarrow g = \rho dx/dt \rightarrow v_{term}= g/\rho$$

$\rho$ will depend on the surface area of the BB, and the nature of it's interation with air, but wouldn't be too hard to measure.
Now, here's the thing. If the BB were dropping in a gravitational field in a vacuum, it's potential energy gets converted into kinetic energy. In this case, since it's velocity isn't increasing, it's potential energy as it falls gets converted into heat.

$$\delta U = gm \delta h = gmv_{term} \delta t \rightarrow \frac{\delta U}{\delta t} = gmv_{term}$$

Some of this heat will go into the air, some into the BB. I don't know how much heat it takes to turn a BB into a "poof of dust", but it happens to debris entering the Earth's atmosphere all the time. The velocities involved there are a lot higher, though. Without knowing more about the melting/poofing temperature of a BB it's difficult to say for certain, but it's not outside the realm of possibilities.

Last edited:
krab
RbrtPtikLeoSeny said:
Am I missing something?
No, your instincts are correct, you friend's are not. If you fire a BB too fast, the air resistance will melt it. But that would be a lot faster than 1800fps.

And, Botemp, the air friction is proportional to the _square_ of velocity.

Correct me if I am wrong but terminal velocity has to deal with falling does it not?

There is no limit to the speed at which something can travel as long as you have the force to give to it (Other then C of course, and as long as it can survive the friction). Only to which an object can fall because of a constant force - ie. gravity.

If say I was pushing a bullet along with my finger at a constant force there would be a point in time at which the bullet no longer accelerates due to friction of the air equalling the force at which I put into the bullet with my finger. But shooting a gun is an entirely different thing, the force stops acting on the bullet extremely quickly, even before it leaves the barrel and friction from the air starts slowing it down at that point in time too, there is no terminal velocity.

I totally agree with Gelsamel. Sure, you can talk about the bullet reaching terminal velocity in the VERTICAL component, but in the HORIZONTAL component of the velocity, the only force acting after the "bang" is the air resistance, and hence there is no equilibration of horizontal forces until the bullet actually stops moving (no velocity, no air resistance).

krab