# Final velocity of an air rifle pellet from compressed gas

• szopaw
In summary, the energy that a pellet propelled by compressed gas from a tank can generate is 6.3 Joules.
szopaw
As in the title, I'm trying to establish the approximate velocity (sans friction and other losses) of a pellet propelled by compressed gas from a tank. Below is what I have came up with myself, I would appreciate if someone could review this as say whether the end values are reasonable.

I have looked for a formula that would give me the energy of the pellet (for which I know the mass and diameter), and I've found the formula for work done in volume on the wikipedia page for the adiabatic process, which goes like this:

W=P1 * V1^γ * ( V2^(1-γ) - V1^(1-γ) ) / (1-γ)

Where P1 is the pressure (in Pascals) in the tank of volume V1 (cubic meters), V2 is the total volume after work has been done (so tank+barrel), γ is the heat capacity ratio (~1.4 for air for this purpose).

Assuming that all of the work is put towards the kinetic energy of the pellet (please tell me if the assumption is unreasonable), that would make the result be

v = sqrt(2*W/m)

Where W is the work from the previous equation and m is the mass (kg) of the pellet

After that, I've run a calculation for what reasonably could be a real air gun with the following values, for a simple scenario:

The starting pressure is 10 bar = 1000000 Pa

Volume V1 of the tank is half volume of the barrel. The barrel has the diameter of the pellet and length of 0.5m.

The pellet has a diameter of 6mm and mass of 0.25 g (0.00025 kg)

So that's V1 ~ 0.00000706858 m3 (~7068.58 mm3)

V2 is the barrel+tank, which is 2 * V1 + V1 = 3 * V1

For γ = 1.4, that gives us
V1^γ ~ 6.15 *10^-8
V2^(1-γ) - V1^(1-γ) ~ 74.032 - 114.886 = -40.866

Therefore

W ~ 10^6 * 6.15*10^-8 * -40.866 / -0.4 ~ 6.3 J

From that

v = sqrt(2 * 6.3 / 0.00025) = sqrt(50400) ~ 224.5 m/s

Does that seem like a number that's anywhere close to what should be true for a 6mm 0.25 g plastic BB pellet shot with 10 bar? Are there any considerable errors in either the assumptions or calculations?

I assume that your 10 bar is gauge pressure. There is 1 atm acting against motion on the barrel side of the bb as it moves through the barrel. I think that air resistance will play a role. You may find the following sites (below) interesting.
The one from NASA has some cool animations showing how air flows around a sphere. But inside the barrel, air does not get to flow around the BB, it is getting shoved out the barrel, rather than flowing around the sphere.
If you look at the chart that i found on Crosman, there are several air rifle velocities much higher than the 225 m/s that you calculated, but these are with lead pellets. So I think with a plastic pellet, it won't take as much energy to accelerate it, but I think you should expect the actual muzzle velocity to be less than your non-friction calculations. How much less? I'm not sure.

https://www.grc.nasa.gov/www/k-12/airplane/dragsphere.html
http://hyperphysics.phy-astr.gsu.edu/hbase/airfri.html
http://www.crosman.com/pdf/CrosmanAirgunPelletCapabilitiesChart20120713.pdf

scottdave said:
I assume that your 10 bar is gauge pressure. There is 1 atm acting against motion on the barrel side of the bb as it moves through the barrel. I think that air resistance will play a role. You may find the following sites (below) interesting.
The one from NASA has some cool animations showing how air flows around a sphere. But inside the barrel, air does not get to flow around the BB, it is getting shoved out the barrel, rather than flowing around the sphere.
If you look at the chart that i found on Crosman, there are several air rifle velocities much higher than the 225 m/s that you calculated, but these are with lead pellets. So I think with a plastic pellet, it won't take as much energy to accelerate it, but I think you should expect the actual muzzle velocity to be less than your non-friction calculations. How much less? I'm not sure.

https://www.grc.nasa.gov/www/k-12/airplane/dragsphere.html
http://hyperphysics.phy-astr.gsu.edu/hbase/airfri.html
http://www.crosman.com/pdf/CrosmanAirgunPelletCapabilitiesChart20120713.pdf

Hey, thank you for those.

Yes, I'm considering gauge pressure and yes, I agree, surface area compared to it's mass dosn't do a plastic BB any favours. Certainly not compared to the drag/mass of a lead pellet.

However my main question is still rather about the adiabatic work and whether 10 bar gauge pressire could reasonably bestow 6 joules of energy on a plastic bb (or appropriately less with drag considered), or are my assumptions off.

The lead pellets (at same velocity) have more kinetic energy, because of higher mass. The amount of energy output looks reasonable. Just that some of it is going to overcome friction. My guess is not a very high percentage, but noticeable.

scottdave said:
The lead pellets (at same velocity) have more kinetic energy, because of higher mass. The amount of energy output looks reasonable. Just that some of it is going to overcome friction. My guess is not a very high percentage, but noticeable.

All right, that's brilliant to hear. I'm not trying to aim for a specific velocity, that's why I'm not concerned about drag or exact values, but I was worried whether the pressure/energy could have been off by an order of magnitude.

scottdave

## 1. What factors affect the final velocity of an air rifle pellet from compressed gas?

The final velocity of an air rifle pellet from compressed gas is affected by several factors, including the type of gas used, the pressure at which the gas is compressed, the weight and shape of the pellet, and the length and diameter of the barrel. Other factors such as temperature and humidity can also have a minor impact on the final velocity.

## 2. How does the type of gas used affect the final velocity of an air rifle pellet?

The type of gas used in an air rifle can have a significant impact on the final velocity of the pellet. CO2 gas, for example, has a lower pressure and density compared to compressed air, resulting in a lower final velocity. On the other hand, compressed air can achieve higher pressures and densities, leading to a higher final velocity.

## 3. Does the weight and shape of the pellet affect its final velocity?

Yes, the weight and shape of the pellet play a crucial role in determining its final velocity. A heavier pellet will require more force to accelerate, resulting in a lower final velocity. Similarly, the shape of the pellet can also affect its aerodynamics, which can impact the final velocity.

## 4. How does the length and diameter of the barrel affect the final velocity of an air rifle pellet?

The length and diameter of the barrel can affect the final velocity of an air rifle pellet in several ways. A longer barrel allows for a longer acceleration time, resulting in a higher final velocity. Similarly, a wider barrel can reduce friction and allow for a smoother passage of the pellet, resulting in a higher final velocity.

## 5. Can the final velocity of an air rifle pellet be accurately predicted?

While there are various mathematical models and formulas that can help predict the final velocity of an air rifle pellet, it is challenging to achieve a completely accurate prediction. This is because there are many factors at play, and small variations in these factors can significantly impact the final velocity. Experimental testing is usually the most reliable way to determine the final velocity of an air rifle pellet.

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