# Free fall velocity with air resistance/drag

• RJLiberator
In summary, the conversation revolves around analyzing free fall motion with air resistance in a computer code class. The topic of air resistance is new to the speaker and they have been searching for information online. The teacher suggested using a decay problem as an example code, but the speaker is struggling to understand how it relates. The equation for drag force and different types of air resistance equations are discussed. The speaker is trying to code a free fall problem with air resistance, but is unsure of how to incorporate it into their code. The conversation also touches on the value of b (a constant in the equation for drag force) and the concept of terminal velocity. It is determined that letting b be user inputted would be the best approach.
RJLiberator
Gold Member

## Homework Statement

I am analyzing free fall motion in my computer code class.
We haven't really discussed much about air resistance and it is a bit of a foreign topic for me.
I am searching the internet as much as I can for information on it, but would really appreciate talking to someone here regarding it.

What I've been giving is
F_d = -bv where b is a constant and v is velocity.
Teacher said to use our decay problem as an example code for this. Suggesting that I may use something of the form Ae^...

I'm just not sure how this all relates. I search online and it seems like there are many many different types/examples of air resistance equations/examples.

So I'm observing
v(t) = mg/b-(e^(-bt))*mg/b
But, in my problem there is no mass.

Sorry to bump, this will be the last time.

I have to code this guy into a free fall problem, problem is, I haven't really worked with air resistance.

Many of the sites (youtube videos and the like) all have the drag force dealing with mass and other such things that I do not consider in my coding problem.

The teacher only gave us this problem:

dv/dt = a - bv

I tried solving this with my limited knowledge of differential equations and found
a = e^(-bt)+bv
But this doesn't make much sense to me as, how can a be a part of the velocity graph.
If acceleration mean the change in velocity maybe I could look at it as a time step? But then e^(-bt) is dependent on time anyway.

Show us how you solved the differential equation, please.

RJLiberator
Will do.

dv/dt = a - bv

Divide across by dv

1/dt = (a-bv)/dv
flip everything (inverse)
dt = dv/(a-bv)

integrate
t = -log(a-bv)/b

Multiply both sides by -b
-bt = log(a-bv)

exponentiate
e^(-bt) = a-bv

a = e^(-bt)+bv

RJLiberator said:
Will do.

dv/dt = a - bv

Divide across by dv

1/dt = (a-bv)/dv
flip everything (inverse)
dt = dv/(a-bv)

integrate
t = -log(a-bv)/b

Multiply both sides by -b
-bt = log(a-bv)

exponentiate
e^(-bt) = a-bv

a = e^(-bt)+bv
You left out the constant of integration, required to satisfy the initial condition at t = 0.

RJLiberator
At time = 0 there would be an initial velocity?

So would it just become a = e^(-bt)+bv-v_i

RJLiberator said:
At time = 0 there would be an initial velocity?

So would it just become a = e^(-bt)+bv-v_i
The initial velocity is zero, and your equation doesn't satisfy that. The correct solution should be:

$$v=a\frac{(1-e^{-bt})}{b}$$

RJLiberator
v = (a-ae^(-bt))/b

So this equation is not dependent on mass, which is good.
But, the problem I am having with this is it is dependent on acceleration and velocity.
I have the duty of coding a graph of the equation of velocity from a free falling object.
I have successfully made the graph for the case without air resistance.

So I figure the graph with air resistance shouldn't be too hard. It should be a simple extension.
When I look at this equation, I am left wondering how to incorporate it into my code.

v is what we are looking for.

1) b, I have no idea what b is. I know it's a constant that is dependent on the fluid it traverses, but how am I suppose to give it numerical value? Should I make it a user input and suggest some values for b?
b is determined by vterm = mg/b
But I don't have a mass, and what would be my terminal velocity? Don't have that either.

I suppose I can let the user input values for b, but I am struggling even finding realistic values to suggest.

2) And then there is acceleration. Can I represent that by a small change in time of the velocity? OR could that Just be gravity = 9.81?

I don't know how you want to handle b. It depends on how technically correct you want to be. Is this for a physics course or a math course?
You should have a = g in your equations. As far as terminal velocity is concerned, it is the value of v when t is infinite. What does your equation say?

RJLiberator
I don't know how you want to handle b. It depends on how technically correct you want to be. Is this for a physics course or a math course?
You should have a = g in your equations. As far as terminal velocity is concerned, it is the value of v when t is infinite. What does your equation say?

I think the best route to go is to let b be user inputted at this point.

Thank you for the confirmation on a = g and understanding of terminal velocity.
When t is infinite, terminal velocity is a/b which is 9.81/b.

This is for a physics course entitled Mathematical and Computational methods for Physicists.

Your help here has been extremely important in my understanding of what I have to do.

## 1. What is free fall velocity with air resistance/drag?

Free fall velocity with air resistance/drag is the speed at which an object falls towards the ground when both gravity and air resistance are acting on it. It takes into account the resistance or drag caused by the air molecules as the object moves through the air.

## 2. How is free fall velocity with air resistance/drag calculated?

The formula for calculating free fall velocity with air resistance/drag is given by: v = √(2mg/k), where v is the velocity, m is the mass of the object, g is the acceleration due to gravity, and k is the drag coefficient. It is important to note that this formula is only an approximation and may vary depending on the shape and size of the object.

## 3. What factors affect free fall velocity with air resistance/drag?

There are several factors that can affect free fall velocity with air resistance/drag. These include the mass and shape of the object, the drag coefficient of the object, the density of the air, and the acceleration due to gravity. Higher mass and larger drag coefficients will result in a lower free fall velocity, while higher air density and gravity will increase the velocity.

## 4. How does air resistance/drag affect the motion of an object in free fall?

Air resistance/drag can significantly affect the motion of an object in free fall. As the object moves through the air, the air molecules exert a force in the opposite direction of the motion, which slows down the object. This means that the object will not continue to accelerate at a constant rate and will eventually reach a maximum falling speed, known as the terminal velocity.

## 5. How can free fall velocity with air resistance/drag be reduced?

The best way to reduce free fall velocity with air resistance/drag is to reduce the drag coefficient of the object. This can be achieved by making the object more streamlined or by using materials that are less resistant to air. Additionally, increasing the mass of the object or decreasing the air density can also help to reduce the free fall velocity. However, the acceleration due to gravity cannot be altered and will always play a role in determining the velocity of an object in free fall.

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