Help needed on exponential decay

In summary, the conversation discusses a research on the relationship between the height level of water in a barrel and the time it takes for the water to flow through a small hole. The researcher is trying to prove that this relationship follows an exponential function, which can be expressed as dh/dt= -kh. In order to do so, they need to establish that the rate of change of volume remaining in the barrel is proportional to the pressure, which can be written as dV/dt = -kp. The researcher is seeking help and suggestions from others to move forward with their research.
  • #1
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.
 
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  • #3


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.
 

1. What is exponential decay?

Exponential decay is a mathematical model that describes the decrease in value of a quantity over time, where the rate of decrease is proportional to the current value.

2. How is exponential decay different from linear decay?

Exponential decay shows a rapid decrease in value at the beginning, followed by a gradual decrease over time. In contrast, linear decay shows a consistent decrease in value over time at a constant rate.

3. What are some real-life examples of exponential decay?

Some examples include the decay of radioactive materials, the cooling of a hot object, and the decrease in population growth rate of a species as it reaches carrying capacity in its environment.

4. How is the decay constant related to the half-life in exponential decay?

The decay constant is the rate at which the quantity is decreasing. It is inversely proportional to the half-life, which is the time it takes for the quantity to decrease by half. The smaller the decay constant, the longer the half-life.

5. How can exponential decay be used in practical applications?

Exponential decay can be used to model and predict the decay of radioactive materials, the depreciation of assets, and the spread of diseases. It can also be used in finance and economics to model interest rates and stock market fluctuations.

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