Raindrop Physics: Solving for Terminal Speed of Falling Drop

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In summary, the raindrop will continue to move at the same speed, and will reach a terminal speed that is effectively constant.
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S0C0M988
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A raindrop of initial mass Mo starts falling from rest under the influence of gravity. Assume that the drop gains mass at a rate proportional to the product of its instantaneous mass and its instantaneous velocity:

dM/dt = kMV

where k is a constant. Show that the speed of the drop eventually becomes effectively constant, and give an expression for the terminal speed. Neglect air resistance.
 
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  • #2
S0C0M988 said:
A raindrop of initial mass Mo starts falling from rest under the influence of gravity. Assume that the drop gains mass at a rate proportional to the product of its instantaneous mass and its instantaneous velocity:

dM/dt = kMV

where k is a constant. Show that the speed of the drop eventually becomes effectively constant, and give an expression for the terminal speed. Neglect air resistance.
This is a poorly worded problem. Mass does not suddenly appear out of nowhere. The resulting motion depends on how the drop is acquring the mass. If it is simply condensing other tiny drops that are moving along with it, that is a completely different situation than if it is running into tiny drops that are at rest. I believe the problem is intended to be treated as the raindrop acquiring additional mass from tiny drops that may be considered at rest until bombarded by the larger drop.

Momentum is conserved in every collision. In this problem things are sticking togeter. See what you can do to set up the problem.
 
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I have no idea how to even start to set up this problem.
 
  • #4
S0C0M988 said:
I have no idea how to even start to set up this problem.
If a raindrop of mass M moving with speed V runs into a bit of water of mass dM, initially at rest, and the two things stick together, how fast will they be moving after the collision assuming no other forces are acting?

It may be that your text is expecting you to assume the validity of the idea that
F = dp/dt = d(MV)/dt with M and V both function of time and do a relatively simple calculus problem. It is a much simpler approach than resorting to momentum conservation fundamentals, but as I stated earlier the assumptions make a difference. You can do the simple calculus problem and hope it is justified, or the more complete problem and justify the result. Choose your approach.
 

1. What is the raindrop physics problem?

The raindrop physics problem is a mathematical and scientific issue that involves understanding the shape, size, and behavior of a falling raindrop. It is a complex problem that has been studied by scientists for many years.

2. How is the shape of a raindrop determined?

The shape of a raindrop is determined by the forces acting upon it as it falls through the air. These forces include air resistance, gravity, and surface tension. As a raindrop falls, it will try to minimize its surface area to reduce the effects of air resistance, resulting in a spherical shape.

3. What factors affect the size of a raindrop?

The size of a raindrop is influenced by several factors, including the temperature and humidity of the air, the altitude at which the raindrop forms, and the duration of its fall. Generally, warmer and more humid conditions will result in larger raindrops.

4. Why do raindrops sometimes appear to merge together as they fall?

When raindrops fall through the air, they can collide and combine with other raindrops, resulting in larger drops. This process is known as coalescence and is influenced by factors such as the speed, size, and shape of the raindrops.

5. How does the size and shape of a raindrop affect its impact on the ground?

The size and shape of a raindrop can greatly impact its impact on the ground. Smaller and more spherical raindrops will have less impact force compared to larger and more irregularly-shaped raindrops. This is due to the relationship between the raindrop's mass and surface area, as well as its terminal velocity.

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