# Showing the Terminal Velocity equation is dimensionally correct.

• Plebert
In summary, the conversation discusses a first year university physics assignment that involves proving the dimensionality of an equation for terminal velocity. The equation involves variables such as mass, density, cross-sectional area, and a dimensionless drag coefficient. The individual discussing the problem struggles with understanding the concept of dimensions and makes an attempt to solve it by substituting variables with their corresponding units, but makes an error in the process. They seek clarification and confirm if they are on the right track.
Plebert
Hey guys, this is my er...first post.
It's a first year university physics assignment that I'm having a bit of trouble with, any help will be rewarded with kind words!(bit of an empty gift, but it's all I have)

Ok, here's the problem.

The terminal velocity of a mass m, moving at ‘high speeds’ through a fluid of density ρ (kg m^3), is given by
V(terminal)=√((2mg)/(DρA))

where A is the cross-sectional area of the object (m2) and D is a dimensionless “drag coefficient”.
Show that equation is dimensionally correct.

Now, not really being certain what the question is asking for regards 'dimensions' hasn't helped but! I did make an attempt by substituting each variable with it's corresponding units.
e.g.

2mg= 2((m/s^2)x(kg))=((m x kg)/ s^2)and ρA=((Kg/m^3)x(m^2))=Kg x m^(-1)

which yields V(ter)=√((mKg)/ s^2)/mKg
=√(s^2) x D
=s x D

This seems more or less nonsensical.
I'm sure it's probably mathematical error or just a failure to grasp the concept of proving an equations dimensions.

Am I wrong?
what is going on?

Plebert said:
Hey guys, this is my er...first post.
It's a first year university physics assignment that I'm having a bit of trouble with, any help will be rewarded with kind words!(bit of an empty gift, but it's all I have)

Ok, here's the problem.

The terminal velocity of a mass m, moving at ‘high speeds’ through a fluid of density ρ (kg m^3), is given by
V(terminal)=√((2mg)/(DρA))

where A is the cross-sectional area of the object (m2) and D is a dimensionless “drag coefficient”.
Show that equation is dimensionally correct.

Now, not really being certain what the question is asking for regards 'dimensions' hasn't helped but! I did make an attempt by substituting each variable with it's corresponding units.
e.g.

2mg= 2((m/s^2)x(kg))=((m x kg)/ s^2)and ρA=((Kg/m^3)x(m^2))=Kg x m^(-1)
Fine up to here. Your dimensions for ρA are therefore kg/m, right?

which yields V(ter)=√((mKg)/ s^2)/mKg
You used kg m instead of kg/m for the dimensions of ρA.

=√(s^2) x D
=s x D

This seems more or less nonsensical.
I'm sure it's probably mathematical error or just a failure to grasp the concept of proving an equations dimensions.

Am I wrong?
what is going on?

## 1. What is the Terminal Velocity equation?

The Terminal Velocity equation is a mathematical formula that calculates the maximum velocity of a falling object when the force of gravity is balanced by the drag force of the surrounding fluid.

## 2. How is the Terminal Velocity equation derived?

The Terminal Velocity equation is derived by equating the forces acting on a falling object, namely the force of gravity and the drag force from the fluid. By setting these two forces equal, we can solve for the maximum velocity at which the object will fall.

## 3. Why is it important to show that the Terminal Velocity equation is dimensionally correct?

Showing that the Terminal Velocity equation is dimensionally correct is important because it ensures that the equation is mathematically sound and that the units for each variable are consistent. This is crucial for the equation to be used in real-world applications and experiments.

## 4. What are the units for the variables in the Terminal Velocity equation?

The units for the variables in the Terminal Velocity equation are: mass (kg), acceleration due to gravity (m/s^2), drag coefficient (unitless), cross-sectional area (m^2), fluid density (kg/m^3), and velocity (m/s).

## 5. Can the Terminal Velocity equation be applied to objects of any shape?

Yes, the Terminal Velocity equation can be applied to objects of any shape as long as the drag coefficient and cross-sectional area are properly calculated for that specific shape. However, it is most commonly used for spherical objects due to their symmetrical shape and ease of calculating the drag coefficient.

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