Showing Constant Acceleration for Falling Raindrop

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In summary, the conversation discusses a problem from classical mechanics involving a falling raindrop that collects mass as it falls. The question is to show that the raindrop has a constant acceleration of g/7 if it starts from rest with a negligible radius. The conversation also includes a discussion of the differential equations needed to solve the problem.
  • #1
blalien
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[SOLVED] Falling raindrop

Homework Statement


This problem is from Gregory. Yeah, classical mechanics is not being kind to me this week. I swear I'm not coming here for every single problem. :-p

A raindrop falls vertically through stationary mist, collecting mass as it falls. The raindrop remains spherical and the rate of mass accretion is proportional to its speed and the square of its radius. Show that, if the drop starts from rest with a negligible radius, then it has constant acceleration g/7.

m: the mass of the raindrop
r: the radius of the raindrop
v: the velocity of the raindrop

Homework Equations


Because the raindrop is a sphere: [tex]m = \rho 4/3 \pi r^3[/tex] for some constant [tex]\rho[/tex]
[tex]m'(t) = k v r^2[/tex] for some constant [tex]k[/tex]

The Attempt at a Solution


Plugging in our equation for m, we have the differential equation
[tex]m' = k v (\frac{3m}{4\pi \rho})^{2/3}[/tex]
With initial conditions m = 0 (or close enough) and v = 0.
We need another differential equation, since both m and v are nonconstant.
Thinking about this intuitively, I don't see why the raindrop isn't just falling with acceleration g. I mean, they're just particles falling, with no other external force. But since I apparently can't assume that the acceleration is g, I see no logical reason to assume that the acceleration is even constant.
So I guess my question is, how can I find the second differential equation?
 
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  • #2
I really, really don't want to sound impatient, because what you guys do is absolutely incredible. But, this homework is due on Tuesday, so I would really appreciate some help. Thanks!
 
  • #3
In the case where the mass isn't constant you need to use F=(d/dt)(mv). That only simplifies to F=ma when the mass is constant. Does that help?
 
  • #4
Use the superposition principle. Separate the action of gravity and raindrop growth. Gravity is easy to deal with. Now, what would happen to the raindrop if there gravity were not present? Conservation of momentum dictates that the raindrop slow down as it accumulates mass. The net behavior of the raindrop is the superposition of the constant acceleration due to gravity and the acceleration (deceleration) due to accumulating mass.
 
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  • #5
> We need another differential equation, since both m and v are nonconstant.

Suppose the drop falls a dist of dx in time dt, and its mass changes from m to m+dm, and velo from v to v+dv. Then,

mv^2/2 + mg*dx = (m+dm)(v+dv)^2/2 =>
mg*dx = mv*dv + v^2*dm/2 =>
mgv = mv dv/dt + (v^2/2) dm/dt.

The v cancels out, giving you your 2nd diff eqn.
 
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Related to Showing Constant Acceleration for Falling Raindrop

1. How can you show constant acceleration for a falling raindrop?

To show constant acceleration for a falling raindrop, you can perform a simple experiment where you drop a raindrop from a certain height and record its position at regular time intervals. By plotting the position versus time data on a graph, you should see a linear relationship, indicating constant acceleration.

2. What is the value of the constant acceleration for a falling raindrop?

The value of the constant acceleration for a falling raindrop is approximately 9.8 meters per second squared. This is due to the force of gravity acting on the raindrop, causing it to accelerate towards the ground at a constant rate.

3. How does air resistance affect the acceleration of a falling raindrop?

Air resistance does have an effect on the acceleration of a falling raindrop. As the raindrop falls, it encounters air resistance which increases as its velocity increases. This results in a decrease in the acceleration of the raindrop, causing it to eventually reach a terminal velocity where the force of air resistance is equal to the force of gravity.

4. Can other factors besides gravity cause acceleration in a falling raindrop?

Yes, other factors such as wind and air currents can also affect the acceleration of a falling raindrop. Wind can cause the raindrop to change its direction and speed, resulting in a change in its acceleration. Additionally, if the raindrop is falling through a cloud, it may encounter updrafts or downdrafts which can also affect its acceleration.

5. How is the concept of constant acceleration important in understanding the motion of a falling raindrop?

The concept of constant acceleration is important in understanding the motion of a falling raindrop as it allows us to predict and analyze the trajectory of the raindrop. By knowing the rate at which the raindrop is accelerating, we can determine its position and velocity at any given time. This concept is also applicable to other objects in motion and is a fundamental principle in the study of mechanics.

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