Dimensions of Air Drag Constants and Terminal Speed Equation

[mdavies23]

Homework Statement

The object is falling vertically in a strange fluid, the magnitude of the air drag is best described by the following F_D = bv+cv² where v is the speed of the object and b and c are constants.
A. What are the dimensions of b and c
B. If the object has mass m find an algebraic expression for the terminal speed V_T in terms of b,c,m, and g

Homework Equations

V = sqrt ( (2 * W) / (Cd * r * g)

The Attempt at a Solution

[F_D] = [v]+[c][v²]
[ML/T²] = [L/T]+[c][L²/T²]

 

[haruspex]

Homework Statement

The object is falling vertically in a strange fluid, the magnitude of the air drag is best described by the following F_D = bv+cv² where v is the speed of the object and b and c are constants.
A. What are the dimensions of b and c
B. If the object has mass m find an algebraic expression for the terminal speed V_T in terms of b,c,m, and g

Homework Equations

V = sqrt ( (2 * W) / (Cd * r * g)

The Attempt at a Solution

[F_D] = [v]+[c][v²]
[ML/T²] = [L/T]+[c][L/T²]

You left out b.
What dimensional rule applies to addition and subtraction of entities?

 

[Marc Rindermann]

and additionally to leaving out b the dimension of $v^2$ is $\frac{L^2}{T^2}$

 

[mdavies23]

You left out b.
What dimensional rule applies to addition and subtraction of entities?

They are equal

 

[haruspex]

They are equal

The dimensionalities are equal, yes. So apply that to the last eqn in post #1, after making Marc's correction in post #3.

 

[mdavies23]

The dimensionalities are equal, yes. So apply that to the last eqn in post #1, after making Marc's correction in post #3.

[ML/T²] =b[L/T]=[c][L²/T²]

 

[mdavies23]

[ML/T²] =b[L/T]=[c][L²/T²]

so I would need an M/T for b and sqrt(L) on top for c

 

[haruspex]

M/T for b

Yes.

sqrt(L) on top for c

How do you get that?

 

[mdavies23]

Yes.

How do you get that?

i mean 1/L

 

[haruspex]

i mean 1/L

Better, but not quite there.
How are you deducing your answers? The simplest is to just write it out as an algebraic equation and simplify: ML/T²=cL²/T².

 

[mdavies23]

Better, but not quite there.
How are you deducing your answers? The simplest is to just write it out as an algebraic equation and simplify: ML/T²=cL²/T².

Oh ok M/L

 

[haruspex]

Oh ok M/L

Right

 

[mdavies23]

Right

so then i can just solve for v correct?

 

[haruspex]

so then i can just solve for v correct?

Dimensional analysis only tells you how the result varies in proportion to the parameters. It does not tell you about any multiplicative constant.

 

[mdavies23]

Dimensional analysis only tells you how the result varies in proportion to the parameters. It does not tell you about any multiplicative constant.

How would i do part b then?

 

[haruspex]

How would i do part b then?

By considering the balance of forces at terminal velocity.