Raindrop Terminal Velocity: Why Do Larger Drops Fall Faster?

[tommyjohn]

Homework Statement

Why do large raindrops fall faster than smaller ones? Find out the mathematical relationship between raindrops terminal velocity and size.
Hint: Assume that the drops are spherical and there density (d=m/v) is constant

Homework Equations

D=m/v

The Attempt at a Solution

I always thought that they would reach a terminal velocity and that they would hit the ground at the same time but apparently I am wrong and want to know why? Thanks in advance

 

[tiny-tim]

Hi tommyjohn!

Hint: the two forces are gravitational and air resistance.

Gravitational force is proportional to …

and air resistance is proportional to … ?

 

[tommyjohn]

gravitational force is proportional to the mass of the object

air resistance is proportional to the square of the speed? ( i had to google it)

i know the gravitational pull is 9.8m/s

so I am assuming the heavier rain drop will fall faster since its mass is larger and due to gravitational force it should fall a little quicker. They both have the same air resistance so its just a matter that the heaver rain drop is heavier.

Is this right or I am just talking bs? lol

 

[tiny-tim]

gravitational force is proportional to the mass of the object

air resistance is proportional to the square of the speed? ( i had to google it)

For two different objects at the same speed, what is it proportional to?

 

[tommyjohn]

if there going the same speed since the gf is dependent it would make the bigger one fall fast since they both have the same air resistance.

 

[RoyalCat]

if there going the same speed since the gf is dependent it would make the bigger one fall fast since they both have the same air resistance.

Galileo would beg to differ..

Think of what happens when you throw a crumpled up piece of paper as opposed to a spread-out sheet of paper.

 

[tommyjohn]

in that case i know the paper that's not crumpled would fall slower because there's more air friction since its more spread out.. is it the same concept for the rain drops eventhough there both spheres?

 

[RoyalCat]

Exactly!

The more surface area an object has, the greater the air resistance on it! Now then, what happens when you scale a raindrop up by some factor?
Compare the forces of gravity (Relative to the mass of the object, which is $$m=\rho V$$ where $$\rho$$ is its mass density and $$V$$ is the volume) with the forces of air drag (Relative to the effective surface area of the object)

To help get you started, I suggest you compare two spherical raindrops. One of radius $$r$$ and one of radius $$R$$, $$R>r$$

Also, consider the definition of terminal velocity. It is the velocity where the force of the air drag is just enough to cancel out the force of gravity, so at that velocity, the object travels at a constant velocity (No acceleration)

Now, as for the force of gravity, imagine for a moment, two objects. One of mass $$M$$ and one of mass $$m$$, where one is heavier than another.

Ignore air drag, and release them from above the ground, with no initial velocity. Write out Newton's second law for each, and tell me which of them falls faster (Which has the greater acceleration?)

 

[tommyjohn]

Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.

SO since the mass of the bigger one is obviously heavier, its acceleration would be slower then that of the smaller one which seems the smaller one would land first but the professor said the bigger one would land first.

 

[RoyalCat]

Newton's second law of motion pertains to the behavior of objects for which all existing forces are not balanced. The second law states that the acceleration of an object is dependent upon two variables - the net force acting upon the object and the mass of the object. The acceleration of an object depends directly upon the net force acting upon the object, and inversely upon the mass of the object. As the force acting upon an object is increased, the acceleration of the object is increased. As the mass of an object is increased, the acceleration of the object is decreased.

SO since the mass of the bigger one is obviously heavier, its acceleration would be slower then that of the smaller one which seems the smaller one would land first but the professor said the bigger one would land first.

Let's separate this into two problems first, since you're getting mixed up about gravity.

Draw all the forces acting on an object of mass $$M$$ (Just gravity, ignore all others for now) and do the same for an object of mass $$m$$

What are their accelerations according to NSL?

The second problem is known as a scaling argument.

Look at the two forces acting on the raindrops. The air drag, and the force of gravity.
For the terminal velocity, it holds true that the force of the air drag is equal to the force of gravity.

$$F_{drag}=kSV^2$$
Where k is just some known constant, S is the effective surface area (For a sphere, this would be half of the surface area of the sphere) of the object, and V is the velocity.

Find the relationship of the terminal velocity to the "size" of the raindrop for the small raindrop and the large raindrop.

From there, it'll be obvious which one comes down first.