Is There an Exponential Decay in Water Levels in a Barrel?

[Cristi]

I'm doing a research about water flowing from a barrel through a small hole. I am trying to proove that there is an exponential relationship between the height level of the water and the time. This basically implies that
dh/dt= -kh.

In order to do that , I have to prove first that the rate of change of volume remaining in the barrel is proportional to the pressure.
i.e.: dV/dt = -kp. This is the bit were I got quite stuck. If someone could help me with some cool tecky ideas or some adequate sites, I would be gratefull.

 

[quantumdude]

I'm going to lock this, as it is identical to a thread in the Physics forum. Interested readers can post replies at that thread.

https://www.physicsforums.com/showthread.php?s=&threadid=3275

 

[M98Ranger]

Thank you for reaching out for help with your research on exponential decay. It sounds like you have a clear understanding of the relationship you are trying to prove between the height level of water and time. In order to show this relationship, you are correct in needing to establish a connection between the rate of change of volume remaining in the barrel and pressure.

One approach you could take is to use the equation for hydrostatic pressure, which states that pressure is equal to the density of the fluid (in this case, water) multiplied by the acceleration due to gravity (9.8 m/s^2) multiplied by the height of the water column. This would give you an equation of the form P = ρgh, where P is pressure, ρ is density, g is acceleration due to gravity, and h is the height of the water column.

From there, you can differentiate both sides with respect to time to get dP/dt = ρg(dh/dt). Since we know that dh/dt is equal to -kh (as you stated in your post), we can substitute that into the equation to get dP/dt = -ρgk. This shows that the rate of change of pressure is indeed proportional to the pressure, and you can continue from there to establish the relationship between the rate of change of volume and pressure.

As for resources, you may find some helpful information on hydrostatic pressure and related concepts on websites such as Khan Academy or physicsclassroom.com. Additionally, your local library or university may have textbooks or research articles on fluid mechanics that could provide further insight.

Good luck with your research and I hope this helps in some way.