I did a few derivations to calculations to figure out the velocity of efflux off a pipe which is situated on the piston of a piston cylinder arrangement.

Attached is the attempt...though I have a doubt in it -

An attachment follows...can someone confirm if it's correct?

I don't see what you are trying to do here - I see a lot of words and not a lot of meaning in that pdf to describe a situation that at face value looks very simple (if poorly defined): flow rate is equal to rate of volume displacement of the cylinder for an incompressible fluid. Your force and acceleration and velocity analysis math doesn't look at all meaningful/useful.

With this and your other thread, you seem to be putting a lot of effort into making simple issues more complicated than they really are without adding any meaning. I don't see any point/value to any of this, nor do I understand why you have an attitude about it.

Well it is a simple situation actually, but I don't find computing what I want simple.

Obviously I do not know the rate of volume displacement of the cylinder, that's why I'm computing it with pressure...the rate at which fluid from the arrangement will flow out is a function of the width of the pipe, which is also unknown; and I have to find the velocity with all these unknown things.

I can't help it!...I am that way!

So what I'm trying to do here is compute the velocity of efflux through that pipe without knowing the area of cross section of that pipe nor having any information about the rate at which the cylinder empties, these are my attempts and I really do not find any attitude in it...it appears to me simple scientific derivations which might be wrong...so I'm trying to see what's wrong in it, or get another right derivation...simple.

My answer above was considering a positive displacement (a given displacement) of the piston. More, coming closer to what you were describing...

-The scenario is ill-defined. You can't figure out the flow rate if you don't know the area of the pipe.
-Using an acceleration calculation isn't really correct unless the force is huge and then it becomes more complicated than we can easily deal with here anyway (you have dynamic effects such as compressibility of the fluid, inertia of the piston, etc). The situation will quickly reach a dynamic equilibrium so you should just use a Bernoulli's principle calculation to find the velocity out the pipe.

There is so very much wrong with your math. Start with your first statement: force=p*k. But above that you said p is force. Well if p is force (so is f?) then p*k can't also be force. In fact, force times area isn't anything useful.

The question does not state a positive displacement.

I do not want to figure out the flow rate, I want to figure out the velocity of efflux which is possible without knowing the thickness of the pipe.

Yes, that's wrong...my fault.

The force should have been f*k/A.

Ok, so I'll try it with Bernoulli's principle which I absolutely do not understand or any attempts to understand it results in failure.

All I've known is that as the velocity of the fluid increases, it's pressure decreases, I cannot make relations between this and law of conservation of energy...they say the decrease in pressure of the ideal fluid is used up to increase the K.E. but everyone also agrees with the fact that significant work cannot be done by the pressure stored in an ideal fluid.

So I'll give another few attempts to compute the answer using the block theory, then proceed to calculate using bernaulli's principal, i.e if I succeed understanding it.

Can someone please do something? I'll try and make the question clearer again -

There's a piston cylinder arrangement, and there's a pipe on top of the piston. What I know -

Dimensions of the piston and cylinder arrangement (not the pipe).
Force on the piston and so pressure in the fluid.
Density of the fluid.
Assume ideal fluid and conditions.