How Much Pressure Is Needed to Launch a Marble at 5 m/s?

[Fifty]

I need to make a cannon that fires a marble. It must be accurate and adjustable enough to hit targets of varying distance and through varying obstacles. For propellant, I want to use compressed air, for which I have an apparatus (valve from a bike tire attached to a PVC cylinder roughly fifteen centimetres long. The cylinder is about five centimetres wide. A quick release valve attached to a string will be the trigger mechanism.

The question is, how much pressure should I load the tank up with to shoot a 5.4 g marble at around 5 metres per second?

My first thought was to use energy: E = Volume(V) x Pressure(P), but I am not sure this is the correct way to do this, mainly because I haven't actually learned this equation in physics class, but I came up with it based on unit analysis during chemistry class. I am not sure what assumptions are made by the equation for example.

My best attempt:

Atmospheric pressure in my classroom is about 14.7 PSI (we measured this in our chemistry class just across the hall) and all of the compressed air will be discharged after every shot. Or rather, enough air will leave the cylinder such that the pressure inside the tank is equal to the pressure outside the tank. If the seal is totally air tight and the pressure perfectly equalizes, the Kinetic energy of the marble after leaving the barrel should be equal to the volume of the cylinder times the difference in pressure (initial pressure in tank minus atmospheric pressure).

Is this correct? Again note that I haven't actually learned this formally.

 

[Mentz114]

The gas equation is PV=RT, where R is a constant.

I think the best approach is to use the force equation to get an acceleration and then integrate this over the distance of the barrel.

If the marble has a cross sectional area A cm and the pressure differential is P, then the acceleration is

a = F/m = AP/m. The velocity is then V = (1/2) a t^2

I hope this helps.

 

[Fifty]

The gas equation is PV=RT, where R is a constant.

I think the best approach is to use the force equation to get an acceleration and then integrate this over the distance of the barrel.

I haven't learned any calculus yet.

If the marble has a cross sectional area A cm and the pressure differential is P, then the acceleration is

a = F/m = AP/m. The velocity is then V = (1/2) a t^2

I hope this helps.

But won't the acceleration change as pressure is released? Initially, the air will push with maximum force, but as the volume increases, won't the pressure the air exerts decrease and thus the force the air exerts? This means I can't use linear equations to solve for the final velocity. I get the idea from my teachers that this can be done with calculus, but I have not learned calculus yet.

 

[Mentz114]

I haven't learned any calculus yet.



But won't the acceleration change as pressure is released? Initially, the air will push with maximum force, but as the volume increases, won't the pressure the air exerts decrease and thus the force the air exerts? This means I can't use linear equations to solve for the final velocity. I get the idea from my teachers that this can be done with calculus, but I have not learned calculus yet.

I can see that not having calculus is a difficulty here. You could try getting an approximate result by assuming the pressure falls linearly or quadratically and finding an average. But I can't see how to get a more accurate estimate without integrating. I'll think about it.

Maybe someone else has a better idea ?

 

[Nickie]

Hello,

Your approach using energy is correct. The equation you used, E = V x P, is known as the ideal gas law and is commonly used to calculate the energy of a gas based on its volume and pressure. However, in this case, we can simplify the equation to just E = P, since the volume of the cylinder is constant and the pressure is the only variable we are changing.

To determine the pressure needed to shoot the marble at 5 meters per second, we can use the equation for kinetic energy, E = 1/2 mv^2, where m is the mass of the marble and v is the desired velocity of 5 m/s. Plugging in the values, we get E = 1/2 (5.4 g) (5 m/s)^2 = 135 mJ.

Now, we need to convert this energy into pressure. This can be done by using the ideal gas law again, but rearranging it to solve for pressure (P = E/V). The volume of your cylinder is 5 cm x 15 cm = 75 cm^3 = 0.000075 m^3. So, the pressure needed would be P = (135 mJ)/(0.000075 m^3) = 1.8 MPa.

However, keep in mind that this calculation is for ideal conditions and does not take into account factors such as friction and air resistance. You may need to adjust the pressure slightly during experimentation to achieve the desired velocity and accuracy.

I hope this helps and good luck with your project!