Have you ever wondered what happens when a ball is dropped into a liquid? How does it cause a splash? In this article, we’ll explore the physics behind it and solve the problem mathematically. We’ll also discuss the assumptions made and possible ways to improve the model.
Problem Statement:
Imagine a spherical ball is dropped from a height h, into a liquid. What is the maximum average height of the displaced water? For instance, although one particular drop of water might travel a high distance, the average displacement height of all the displaced water might be very low.
Approximation:
Let’s make some assumptions. Let’s assume that on impact, a fraction λ of the ball’s Kinetic Energy is transferred to the liquid. Let’s also assume that all of the energy transferred will go into shooting water up. This should give the upper bound on how high the water “tower” will be.
The Kinetic Energy of the ball at impact with the surface of the liquid is,
KE = m * g * h
The fraction that is transferred into making the water tower is,
KEt = m * g * λ * h
When the formation of the water tower reaches its maximum height, the potential energy PE must be equal to KEt. We’ll assume that the tower has cross-sectional area A, and integrate the PE of each slice of the tower at a particular height to get the total PE.
PE = ∫0ht g * l * ρ * A * dl = g * ρ * A * ½ * ht2
Where ht is the height of the water tower, ρ is the density of the liquid, and g is gravity.
Set KEt = PE and get,
m * g * λ * h = g * ρ * A * ½ * ht2
We’ll use A = πr2, which assumes the water tower has a radius equal to the ball. Solving for ht, we note that the g terms cancel, and we get.
ht = 1/r * √{(2 * λ * m * h) / (π * ρ)}
Let’s take λ = 1. Let’s use a water droplet. It has a radius of 0.5 cm, a mass of 0.125 g. Let’s drop it into water from a height of 20 cm.
We get,
ht = 2.52... cm
Referring back to the original problem statement, we need only divide ht by two to get the average height of the displaced water.
Thus,
μht = 1.26... cm
Questions
1. Are there any results that point to flaws in my model?
It is important to test any model against real-world data to check for its validity. In this case, we have only tested the model for a water droplet dropped into water. It would be helpful to test the model under different conditions, such as using different types of liquids or dropping objects of different shapes and sizes. This could point to flaws in the assumptions made or the model itself.
2. Given this initial approximation, how does one go about adding in more complex phenomena?
One way to improve the model is to include additional factors that were not previously considered. For example, the model currently assumes that all of the energy transferred goes into shooting water up. In reality, some of the energy is lost due to friction between the ball and the liquid, or due to deformation of the ball upon impact. These factors can be incorporated by modifying the assumptions and equations used.
Another way to improve the model is to conduct experiments to gather data on the phenomenon. For example, one could measure the height of the water tower for different types of liquids, or for different shapes and sizes of objects. This data could be used to update and refine the model.
Conclusion
In conclusion, we have explored the physics behind the splash caused by a ball dropped into a liquid, and solved the problem mathematically. We have discussed the assumptions made and possible ways to improve the model. It is important to test any model against real-world data and update it as necessary to ensure its accuracy and usefulness.
How Does a Ball Cause a Splash? (with the Relevant Math)
Have you ever wondered what happens when a ball is dropped into a liquid? How does it cause a splash? In this article, we’ll explore the physics behind it and solve the problem mathematically. We’ll also discuss the assumptions made and possible ways to improve the model.
Problem Statement:
Imagine a spherical ball is dropped from a height h, into a liquid. What is the maximum average height of the displaced water? For instance, although one particular drop of water might travel a high distance, the average displacement height of all the displaced water might be very low.
Approximation:
Let’s make some assumptions. Let’s assume that on impact, a fraction λ of the ball’s Kinetic Energy is transferred to the liquid. Let’s also assume that all of the energy transferred will go into shooting water up. This should give the upper bound on how high the water “tower” will be.
The Kinetic Energy of the ball at impact with the surface of the liquid is,
The fraction that is transferred into making the water tower is,
When the formation of the water tower reaches its maximum height, the potential energy PE must be equal to KEt. We’ll assume that the tower has cross-sectional area A, and integrate the PE of each slice of the tower at a particular height to get the total PE.
Where ht is the height of the water tower, ρ is the density of the liquid, and g is gravity.
Set KEt = PE and get,
We’ll use A = πr2, which assumes the water tower has a radius equal to the ball. Solving for ht, we note that the g terms cancel, and we get.
ht = 1/r * √{(2 * λ * m * h) / (π * ρ)}Let’s take λ = 1. Let’s use a water droplet. It has a radius of 0.5 cm, a mass of 0.125 g. Let’s drop it into water from a height of 20 cm.
We get,
Referring back to the original problem statement, we need only divide ht by two to get the average height of the displaced water.
Thus,
Questions
1. Are there any results that point to flaws in my model?
It is important to test any model against real-world data to check for its validity. In this case, we have only tested the model for a water droplet dropped into water. It would be helpful to test the model under different conditions, such as using different types of liquids or dropping objects of different shapes and sizes. This could point to flaws in the assumptions made or the model itself.
2. Given this initial approximation, how does one go about adding in more complex phenomena?
One way to improve the model is to include additional factors that were not previously considered. For example, the model currently assumes that all of the energy transferred goes into shooting water up. In reality, some of the energy is lost due to friction between the ball and the liquid, or due to deformation of the ball upon impact. These factors can be incorporated by modifying the assumptions and equations used.
Another way to improve the model is to conduct experiments to gather data on the phenomenon. For example, one could measure the height of the water tower for different types of liquids, or for different shapes and sizes of objects. This data could be used to update and refine the model.
Conclusion
In conclusion, we have explored the physics behind the splash caused by a ball dropped into a liquid, and solved the problem mathematically. We have discussed the assumptions made and possible ways to improve the model. It is important to test any model against real-world data and update it as necessary to ensure its accuracy and usefulness.