BR Physics - Terminal Velocity

[Lunasly]

Q.If a raindrop freezes into a round hailstone, how does its new terminal velocity compare with the terminal velocity it would have had it had remained as a rain drop?

Use the formula:

Vt = terminal velocity,
m = mass of the falling object
g = acceleration due to gravity
C = drag coefficient
ρ = density of the medium of which the object is falling through
A = projected area of the object/cross-sectional area

A) Terminal velocity stays the same
B) Terminal Velocity gets smaller
C) Terminal Velocity gets bigger
D) The relationship depends on how m changes

Answer: B (highlight to see answer)

Alright, in order for terminal velocity to decrease we either have to decrease the value of a variable int he numerator or increase the value of a variable int he denominator. Here is my logic behind all of this: If the rain-drop (which is composed of water) turns into ice, the first thing that pops into my head is that density decreases (as ice is less dense then water). Hence, that is the reason why I picked C. I didn't think the mass would change as it is the same amount of matter. The shape is roughly the same between the two so C doesn't change. Gravity is obviously the same. The one thing that didn't come to my head, and that I found out later, is that ice expands. Which makes sense because the density of water decreases, and so the the cross-sectional area of the hail would be larger then a rain drop. The back of the book says that density doesn't change. Why?

Thanks.

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[Lunasly]

Nevermind, p is the density of the medium, not the object.

 

[nabilesmail]

Maybe this makes sense. When turning into a hail stone, water expands (as it forms ice)= more volume. More volume creates a bigger boyant force pushing up, therefore lowering the speed down?

Not sure how to reason this with the formula though :/

is there another formula also included in the passage?

 

[Lunasly]

Well in regards tot he formula, since the ice expands, it's cross-sectional area expands. Cross-secetional area is represented as the variable 'A' in the equation. 'A' is indirectly proportional to terminal velocity, so it if it increases, then terminal velocity decreases.

 

[Tatiana3325]

Maybe this makes sense. When turning into a hail stone, water expands (as it forms ice)= more volume. More volume creates a bigger boyant force pushing up, therefore lowering the speed down?

Not sure how to reason this with the formula though :/

is there another formula also included in the passage?

Don't think about hail forming. That's enormously complicated, forms in different ways and depends on many, many variables.

In this case, were talking about a perfect sphere. The equation isn't necessary to answer the multiple choice question if the concept is understood.

 

[Lunasly]

What do you mean? The concept here is that ice expands as it forms. This leads us to believe that the area is larger. Area is indirectly proportional. The equation is necessary.