Falling Water Droplet Homework: Show Radius & Speed Increase Linearly over Time

In summary, the conversation discusses a scenario where a water droplet falls through a humid atmosphere and gains mass at a proportional rate to its cross-sectional area. It is assumed that the droplet starts from rest and that its initial radius is small enough to not experience any resistive force. The equation relating area to volume is mentioned, but the speaker is unsure of how to proceed with the problem. They also mention that the question is flawed and suggest a better assumption. The differential equation is then solved to give the first answer needed, and momentum conservation is discussed as a way to find the second answer.
  • #1
Varnson
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Homework Statement


Assume that a water droplet falling through a humid atmosphere gathers up mass at a rate that is proportional to its cross-sectional area A. Assumethat the droplet starts from rest and that its initial radius R0 is so small that it suffers no resistive force. Show that its radius adn its speed increase lineraly with time.


Homework Equations


A= pi[(4/3)pi]^(-2/3)*Vol^(2/3)


The Attempt at a Solution



I found the above equation to relate the area to the volume. I am stuck on this and do not know which direction I should head in.
 
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  • #2
That's a poor question; it's forcing you to make a plainly unphysical assumption, namely that [itex]\dot{m}(t)=km^{2/3}(t) [/itex], when a much better assumption would be [itex] \dot{m}(t)=kv(t)m^{2/3}(t) [/itex]. However, the latter does not produce an easy answer, while the former does.

That said ... the above differential equation is trivially solved to give you the first answer that you need. For the second answer, you need to write an equation for momentum conservation and plug in the result from the first answer.
 
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What is the purpose of the "Falling Water Droplet Homework"?

The purpose of the "Falling Water Droplet Homework" is to understand the relationship between the radius and speed of a falling water droplet over time. By completing this homework, students will learn about the concept of linear increase and how it applies to the motion of a falling droplet.

What does it mean for the radius and speed to increase linearly over time?

When the radius and speed of a falling water droplet increase linearly over time, it means that there is a constant rate of change between these two variables. In other words, as time increases, the radius and speed of the droplet will increase at a steady and predictable rate.

How do you calculate the radius and speed of a falling water droplet?

The radius of a falling water droplet can be calculated by measuring its diameter and dividing it by 2. The speed can be calculated by dividing the distance the droplet falls by the time it takes to fall that distance.

What factors can affect the radius and speed of a falling water droplet?

The main factors that can affect the radius and speed of a falling water droplet are air resistance, gravity, and the initial height of the droplet. Air resistance can slow down the droplet's fall, while gravity pulls it downwards. The initial height of the droplet can also impact its speed and radius, as a higher starting point will give it more time to increase in size and speed before hitting the ground.

How is this homework relevant to real-world applications?

Understanding the relationship between the radius and speed of a falling water droplet is relevant to many real-world applications, such as studying the motion of raindrops or predicting the behavior of liquids in industrial processes. It can also help in understanding the principles of physics and how they apply to everyday phenomena.

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