Dimensions of Air Drag Constants and Terminal Speed Equation

In summary, the object is falling vertically in a strange fluid with air drag described by FD=bv+cv2, where v is the speed of the object and b and c are constants. The dimensions of b and c are M/L and 1/L, respectively. If the object has mass m, the algebraic expression for the terminal speed VT in terms of b, c, m, and g is VT = sqrt((2mg)/(bcg)). This is found by considering the balance of forces at terminal velocity.
  • #1
mdavies23

Homework Statement


The object is falling vertically in a strange fluid, the magnitude of the air drag is best described by the following FD = 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 VT in terms of b,c,m, and g

Homework Equations


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

The Attempt at a Solution


[FD] = [v]+[c][v2]
[ML/T2] = [L/T]+[c][L2/T2]
 
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  • #2
mdavies23 said:

Homework Statement


The object is falling vertically in a strange fluid, the magnitude of the air drag is best described by the following FD = 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 VT in terms of b,c,m, and g

Homework Equations


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

The Attempt at a Solution


[FD] = [v]+[c][v2]
[ML/T2] = [L/T]+[c][L/T2]
You left out b.
What dimensional rule applies to addition and subtraction of entities?
 
  • #3
and additionally to leaving out b the dimension of ##v^2## is ##\frac{L^2}{T^2}##
 
  • #4
haruspex said:
You left out b.
What dimensional rule applies to addition and subtraction of entities?
They are equal
 
  • #5
mdavies23 said:
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.
 
  • #6
haruspex said:
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]
 
  • #7
mdavies23 said:
[ML/T2] =b[L/T]=[c][L2/T2]
so I would need an M/T for b and sqrt(L) on top for c
 
  • #8
mdavies23 said:
M/T for b
Yes.
mdavies23 said:
sqrt(L) on top for c
How do you get that?
 
  • #9
haruspex said:
Yes.

How do you get that?
i mean 1/L
 
  • #10
mdavies23 said:
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.
 
  • #11
haruspex said:
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
 
  • #12
mdavies23 said:
Oh ok M/L
Right
 
  • #13
haruspex said:
Right
so then i can just solve for v correct?
 
  • #14
mdavies23 said:
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.
 
  • #15
haruspex said:
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?
 
  • #16
mdavies23 said:
How would i do part b then?
By considering the balance of forces at terminal velocity.
 

FAQ: Dimensions of Air Drag Constants and Terminal Speed Equation

1. What is air drag and how does it affect objects?

Air drag, also known as air resistance, is the force exerted by air molecules on objects as they move through the air. This force opposes the motion of the object and can cause it to slow down. The amount of air drag depends on the size, shape, and speed of the object, as well as the density and viscosity of the air.

2. How do you calculate the dimensions of air drag?

The dimensions of air drag can be calculated using the following formula: Drag force = Cd * (1/2) * ρ * V^2 * A, where Cd is the drag coefficient, ρ is the density of air, V is the velocity of the object, and A is the cross-sectional area of the object. The drag coefficient is a dimensionless number that represents the object's shape and surface properties.

3. What factors affect the drag coefficient of an object?

The drag coefficient of an object can be affected by its shape, surface texture, and orientation. Objects with a streamlined shape, such as a sphere or teardrop, have a lower drag coefficient compared to objects with a flat or irregular shape. A rough or textured surface also increases the drag coefficient. The orientation of the object relative to the direction of motion can also affect the drag coefficient.

4. How does air density affect air drag?

Air density is a measure of how compact or dense air molecules are in a given volume. The density of air can affect air drag because a higher density means there are more air molecules present to exert a force on the object. This results in a higher drag force, which can slow down the object more quickly. Air density is affected by factors such as altitude, temperature, and humidity.

5. Can air drag be reduced or eliminated?

Air drag can be reduced by changing the shape, surface texture, or orientation of an object. For example, using a streamlined shape or smoothing out the surface of an object can decrease the drag coefficient and therefore reduce the drag force. However, it cannot be completely eliminated as long as an object is moving through the air. In some cases, air drag can be counteracted by other forces, such as lift or thrust, to maintain the motion of the object.

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