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In 5:49 in this MythBusters video https://www.youtube.com/watch?v=74K3IYOM7Lg, the MythBusters Jamie and Adam believe that barrel length is the deciding factor in acceleration, more so than PSI. Why is this? An idea just popped into my head that the longer the ping-pong ball is in the barrel, the more time the air has to facilitate the barrel, thus pushing the ball. But assuming a diameter of 0.04 meters and a weight of 0.0028 kg, at what point does the ball accelerate with less than 0.5 m/s^2?

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    $\begingroup$ Mythbusters is a entertaining show, and I'm all for anything that demonstrates the value of the empirical method to all and sundry, but ... the level of analysis they bring to their problems is suited to the general audience at which they target the show. The analysis is rudimentary and purely qualitative. Don't read too much into it. $\endgroup$ Commented Aug 4, 2015 at 2:22
  • $\begingroup$ @dmckee They bring up a good point, however, that barrel length determines acceleration more than PSI does (especially if we assume the barrel is frictionless). It did get me thinking, so I cannot discredit the show here. Also, that is more of an ad hominem attack. Even more so, I don't think they would buy all of that fancy equipment for opinionated science, but I am beginning to digress. $\endgroup$
    – Jossie
    Commented Aug 4, 2015 at 2:31
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    $\begingroup$ I didn't say anything about opinion, I say the analysis is simple and qualitative. There is a big difference. I like Mythbusters and use some clips from the show in my General Education classes. A quantitative analysis would tell you the functional dependence of muzzle velocity on both parameters. Then you would see several important things, like neglecting barrel friction is only viable for high-pressure systems (essentially all modern firearms), but the point is that you would be able to talk about how much change in pressure equates to how much change in length for a particular firearm. $\endgroup$ Commented Aug 4, 2015 at 3:08
  • $\begingroup$ Up to that point they were talking about velocity, not acceleration. I think he just misspoke. It makes perfect sense that the velocity of the projectile will benefit more from increasing the length of the barrel than from increasing the pressure. $\endgroup$ Commented Aug 4, 2015 at 4:10

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The work involved in accelerating the ping-pong ball is the integral of P(dV). If you further use the work-kinetic energy theorem, you arrive at the velocity of the ping-pong ball. Obviously, for the long barrel, this method doesn't work. In my opinion, this is because there is a hidden assumption that the ping-pong ball fits tightly against the tube that it is being shot through, such that you maintain a vacuum all of the way down the length of the barrel that is in front of the ping-pong ball.

The high speed video clearly implies that there is a small amount of air leakage around the edges of the ball. Such air leakage would be expected to travel around the ball at sonic velocity, so some of the air is getting past the ball as it is being accelerated down the tube and before it gets up to its final speed. By the time the ball gets to the end of the long tube, it starts compressing the air that is in front of it (it does work on the air), and as shown, comes to a stop when it compresses the air enough (all of its kinetic energy is turned back into compression work).

I'm not sure if there is a device or method that would guarantee no air leakage around the ball, but if this could be done, it is my opinion that there would be a good correlation between barrel length and exit velocity, up to approximately sonic velocity for the case where atmospheric pressure alone is accelerating the ping-pong ball.

At the end of the video segment, where 300-500 psi air was also applied to the long tube, the result was similar to some of the pumpkin shooters that use compressed air. Even for something as sturdy as a pumpkin, if the pressure pushing the pumpkin down a barrel is extremely high, it will turn the pumpkin into a find "mist" that is dispersed throughout the air stream leaving the barrel, which means that the pumpkin launch failed (it doesn't count in the competition for distance).

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  • $\begingroup$ Why sonic velocity? $\endgroup$
    – Jossie
    Commented Aug 4, 2015 at 14:45
  • $\begingroup$ When the ratio of pressures between pressure outside the pipe to pressure inside the pipe (which is at vacuum) exceeds approximately 2:1 (if memory serves), the flow rate of air into the pipe can only go approximately sonic due to pressure drop effects. A converging/diverging nozzle may be able to obtain a higher tube velocity, but that device wasn't used in the Mythbusters device. $\endgroup$ Commented Aug 4, 2015 at 14:52
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Well, most importantly, this question is fundamentally wrong... The units for acceleration are meters/(second**2). Either you made a typo, or you don't fully understand what you are trying to ask. I will assume it was just a typo.

The first detail that must be addressed is the type of system we are analyzing.

If it is something like a tube attached to an air hose that consistently feeds air into the tube, the pressure and barrel length are both mutually important in an ideal friction-less situation.

In the situation where there is friction it is more important to have higher pressure, because as the ball slides in a longer barrel, it will lose energy to friction by rubbing the inside of the barrel, which is why a shorter barrel with higher pressure is preferred in that arrangement.

Now, let's analyze the limited reagent scenario, such as bullets and vacuum cannons.

Firstly, the thing that goes bang! While there are MANY variables in this situation, we will remove a few by assuming all of the components in the gun are perfectly thermally insulated; meaning that the expanding gas has zero heat transfer to the barrel and other parts of the gun. There would need to be a special balance between barrel length and amount of powder(pressure) that is packed behind the bullet. Also, I will assume that the bullet completely fills the barrel in at least one cross-sectional slice, making a complete separation between the air outside the barrel and air inside the barrel. So we would want a barrel length that allows the gas to expand to point where the pressure inside the barrel is the same as the pressure outside, or at least very near it, and then let the bullet escape. If the barrel is too long, it will hold the bullet past that point and will create a suction due to a difference in pressures inside and outside the barrel, thus slowing the bullet.

I won't answer any of the numbers because they are not quite coherent. Enjoy. Will

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