how would i calculate the velocity coming out of an air cannon in terms of pressure, volume,barrel length, area of barrel opening, k(coffecient of porpotionalityin -kv^2 for friction)?

Draw the free body diagram for the shell being fired, calculate forces acting on it, get an expression for acceleration and integrate to get velocity.

 

You would need a law governing pressure and volume of air inside the pressure barrel, Eg.PV= constt

i wish it was that easy but since i have friction in it becomesmuch harder

 

[math]f_{net}=f_{pressure}+f_{friction}[/math]

[math]f=pa[/math]

[math]f=p_iv_ia/(v_i+ax)[/math] that equals the force from the air cannon

 

using the equation p_iv_i=p_fv_i and since the volume increases by barrel opening(andassuming theprojectile fits snug in the tube) andhow far it has been shotthrough the tube

 

[math]a=p_iv_ia/m(v_i+ax)-kv^2/m[/math]

 

[math]y''=p_iv_ia/m(v_i+ay)-ky'^2/m[/math]

 

but when i solved this diff equation i got obviously wrong equation maybe i just did the math wrong i dont know. i dont even know if that would give the right answer