# Sinking bucket - differential equations

• markswabinski
In summary, the conversation pertains to a question from the course Ordinary Differential Equation on how fast a bucket will sink completely under water when a hole is made underwater. The speaker is having difficulty creating a model for solving this problem, but has attempted to establish an ODE by considering the initial state of the bucket and the rate of filling it. The conversation ends with a request for help or guidance in the right direction.

## Homework Statement

Let's have a bucket flowing in water. Now we make a hole underwater. How fast will the bucket sink completely under water?

It is a question from course called Ordinary Differential Equation, so I'm supposed to establish an ODE to solve this problem. I understand how to solve such equation, but I'm really bad at creating models.

The details about water, where the hole is or how the bucket looks like are not specified - can either be general or taken to be insignificant.

## The Attempt at a Solution

I've tried by specifying the t0 state: we know that it floats, so buoyancy equals gravity: m0 = rho_w * h * Ab (where m0 is the mass of bucket, rho_w is the density of water, h is how much the bucket has sunk when floating and Ab is the area of base of backet assuming rectangular bucket for simplicity).

Now the hole appears. I have no idea how to employ the rate of flowing water (since differentials are to be used) into the equations to come up with an model to solve.

Any help or the nudge towards right direction (articles / solved examples) highly appreciated.

Last edited:
hi markswabinski! welcome to pf!

i assume the bucket starts empty

suppose the rate of filling the bucket is v(t)

what do you think v is likely to be proportional to?

write that as a differential equation (don't bother with any constants such as ρ or A, just include them all in one big constant C ) …

show us what you get

## 1. What is a sinking bucket?

A sinking bucket is a physical model used to study the dynamics of a fluid filling or draining from a container through a small hole at the bottom. It is often used as an example in differential equations to understand the relationship between the variables involved in the process.

## 2. How is a sinking bucket related to differential equations?

The process of fluid filling or draining from a sinking bucket can be described using differential equations. These equations help us understand the changing rates of volume, height, and time in the bucket as the fluid fills or drains out.

## 3. What are the variables involved in the sinking bucket differential equation?

The variables involved in the sinking bucket differential equation are the volume of fluid in the bucket, the height of the fluid, the rate of change of height, and the rate of flow through the hole at the bottom of the bucket. These variables are all interrelated and can be described using differential equations.

## 4. What is the importance of studying the sinking bucket differential equation?

The sinking bucket differential equation is an important model in understanding the dynamics of fluid flow in various systems. It can help scientists and engineers make predictions and solve problems related to fluid mechanics in real-world scenarios.

## 5. What are some applications of the sinking bucket differential equation?

The sinking bucket differential equation has various applications in fields such as engineering, physics, and environmental science. It can be used to study the filling and draining of tanks, the flow of water through pipes, and the movement of groundwater in soil. It is also helpful in understanding the behavior of fluids in natural disasters like floods and tsunamis.