Equating Gas Pressure and Projectile Stagnation Pressure

[user1111999]

Imagine a projectile with density p, initial velocity v0 and initial stagnation pressure = 0.5p{v0}^2 being fired into the gas chamber with a final gas pressure given by Pg = nkT, for n being the final number density before (at max pressure of gas) and T the corresponding temperature of the gas.

Is it true that the initial stagnation pressure and final gas pressure can be related, as well as the final gas energy and initial projectile kinetic energy? Or mathematically are the following statements true?

0.5p{v0}^2 = nkT (Statement 1)
and
0.5m{v0}^2 = NkT

12ρpvp02=nkBTf and12mpvp02=NkBTf For m being the mass of the projectile and N being the number of particles in the gas. I have attached a diagram of the intended thought experiment below:

 

[Chestermiller]

Only a fluid can have a stagnation pressure. A solid projectile can't.

 

[Arjan82]

Imagine a projectile with density p, initial velocity v0 and initial stagnation pressure = 0.5p{v0}^2

the equation $p = 0.5\rho v_0^2$ (now $p$ is pressure and $\rho$ is density, as per usual convention) means nothing in this case. You should use its kinetic energy: $E_k = 0.5m v_0^2$.

being fired into the gas chamber with a final gas pressure given by Pg = nkT, for n being the final number density before (at max pressure of gas) and T the corresponding temperature of the gas.

But what is $g$ in this equation? Usually the symbol is used for gravitational acceleration, but that makes no sense here. And $k$ is the Boltzmann constant?

Also, in common notation regarding the gas law $n$ is the amount of substance (gas) in moles. The term 'number density' means nothing to me.

The equation $Pg=nkT$ is either incorrect or uses strange symbols for common quantities, making it unreadable. Could you specify the actual equation you mean?

Is it true that the initial stagnation pressure and final gas pressure can be related, as well as the final gas energy and initial projectile kinetic energy? Or mathematically are the following statements true?

You can only relate the energies, and actually only energy differences (i.e. the change in kinetic energy is added to the gas). Using the gas law is not enough here. With the gas law you can equate:

$$
\frac{p_1 V_1}{T_1} = \frac{p_2 V_2}{T_2}
$$

with $V$ the volume. But then you need the pressure in the end condition, for which you need something like the isentropic flow relations. But then again, that assumes, well, isentropic flow. Usually slamming a 'projectile' into some container is not isentropic.

(or actually, I'm thinking, maybe the problem is that it does not remain Barotropic? I'm not sure here. But that of course also depends on how fast the projectile is slamming into the container)

12ρpvp02=nkBTf and12mpvp02=NkBTf

This is gibberish to me. Please use the LaTeX Guide.

 

[berkeman]

Thread closed temporarily for Moderation...

 

[berkeman]

At OP's request, the thread will remain closed. Thanks folks.