# Need help with D.E to account for air drag of my projectile

• guipenguin
In summary, the speaker discusses their recent project of creating a pneumatic air cannon and the difficulties they faced in accurately measuring the initial velocity of a projectile. They mention the need for differential equations to account for air resistance and ask for advice on how to set up the equations. They also mention their background in Calculus II and willingness to put in the necessary effort to solve the problem.
guipenguin
This past weekend, for fun, I made a pneumatic air cannon, using compressed air to shoot out a small projectile at very high velocity. I'd post my video, but I have not made 15 posts on this forum.

What I need to do is find as close as I can to the actual initial
velocity of a projectile, once it leaves the barrel. If I have the air
cannon pointed at 90 degrees to the ground, than I could shoot the
projectile vertical and time hang time, factoring in known
acceleration of gravity, to find an approx initial velocity. The only
problem is, with a projectile going this fast, with significant
surface area, there will be significant air drag to account for.

As air resistance effects the velocity of the projectile, and as
decreased velocity coincides with a decrease in drag, I believe I will
be needing to use differential equations.

Do you have any idea how I could setup a differential equation to
account for air resistance in my quest to find as close as I can to
the projectile's initial velocity? I have never done anything with air drag before. I just finished Calculus II, but I can put in the nessesary time and effort if someone can get me started here.

Thanks,
John

How are you modeling the air resistance? Two commonly used ways are "linear";F= -kv, where F is the friction force due to air resistance, v is the velocity vector, and k is a constant, or "quadratic"; F= -k |v|v, where |v| is the 'length' of the velocity vector or speed. This is "quadratic" because |F|= -k|v|2.

The first case is fairly simple: the x coordinate of velocity is given by mvx'= -kvx and the y coordinate of velocity is given by mvy'= -kvy- (1/2)mg. Unfortunately that is only accurate for very low speeds.

The quadratic case is harder: the x coordinate of velocity is given by

$mv_x'= -k\sqrt{v_x^2+ v_y^2}v_x$, $mv_y'= -k\sqrt{v_x^2+ v_y^2}v_y- (1/2)mg$.

## 1. What is D.E and why is it important in accounting for air drag of a projectile?

D.E stands for Differential Equation, which is a mathematical equation that describes the relationship between a function and its derivatives. In the context of projectiles, D.E is important because it allows us to model the complex motion of a projectile in the presence of air drag.

## 2. How does air drag affect the motion of a projectile?

Air drag, also known as air resistance, is a force that opposes the motion of an object through the air. This force increases with the speed of the object and can significantly alter the trajectory and velocity of a projectile. Therefore, it is important to account for air drag in order to accurately predict the motion of a projectile.

## 3. What are the factors that affect the air drag of a projectile?

The air drag of a projectile is affected by several factors, including the speed and direction of the projectile, the shape and size of the projectile, and the properties of the air, such as density and viscosity. These factors can vary depending on the specific situation, so it is important to consider them when accounting for air drag.

## 4. How is D.E used to account for air drag in projectile motion?

To account for air drag in projectile motion, we use a D.E called the drag equation, which relates the air drag force to the velocity of the projectile. By incorporating this equation into the overall equations of motion for the projectile, we can accurately model the trajectory and velocity of the projectile in the presence of air drag.

## 5. Are there any simplifications or assumptions made in accounting for air drag of a projectile?

Yes, there are some simplifications and assumptions made when accounting for air drag in projectile motion. These include assuming a constant air density and neglecting factors such as turbulence and wind. These simplifications allow for a more manageable and solvable D.E, but they may result in slight discrepancies between the predicted and actual motion of the projectile.

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