My question is one I have searched on Google many times but with no success. I am now using Yahoo! Answers as a last resort to have my question answered.
I already know how to use the trajectory WITHOUT air resistance (drag) but I wish to step up my knowledge and use it WITH air resistance.
The equation for without is this: x=vt*cos(θ) and y=vt*sin(θ)-1/2*gt²
My question is, how do I insert into this equation air resistance? If it requires a formula, please show it. It would also be nice if I see a step-by-step example on how to solve it with air resistance.. Example Problem: A cannon has a velocity of 15 m/s shooting at an angle of 62° at a time of 1 second. The wind is not present, so all that affects the cannon ball is the air resistance. Without Air resistance, the equation equals x=7.04 and y=8.33. Now, I just need to know what the equation equals WITH air resistance.
I also wish to know what would happen and how I would calculate if there is wind affecting the cannon ball. Like, what would be the difference in the cannons trajectory if there is two mile an hour wind (either blowing with or against the cannon ball, your choice! I still learn from it either way.) instead of no wind at all?
This is purely based on my interest, for I do love math so!
Thank you for spending your time to help my education!
You will not find an accurate trajectory equation for a system with air resistance, because the equations describing the drag force contain constants which can change depending on temperature, orientation of the object, ect. And it’s difficult to account for these variables, and it can easily result in chaotic behavior.
The equations for the drag force of a fluid are (air is a fluid)
F = -b v + ½ ρ v² A C
where
F is the force of drag,
ρ is the density of the fluid,[3]
v is the speed of the object relative to the fluid,
A is the reference area,
C is the drag coefficient (a dimensionless parameter)
and
b is a constant that depends on the properties of the fluid and the dimensions of the object.
Usually, you can ignore one of these two terms, depending on whether you’re at high or low speeds. At high speeds v² > v ==> ρ v² A C > b v
At low speeds, v > v² ==> b v > ρ v² A C
For a launching a cannon ball, you’ll probably work in the region of high speeds, and can just use the equation
m y″ = -½ ρ A C ( (x′)²+ (y′)² ) – mg
m x″ = -½ ρ A C ( (x′)²+ (y′)² )
The problem in solving these equations is that C, the drag coefficient, isn’t constant. If you assume it’s constant, then you can solve these coupled differential equations to get trajectory equations.
You should look at the you wikipedia link below which does give the equation for a falling object, but not an equation of an object with a non-zero component of velocity on the x-direction.
February 26th, 2010 at 3:59 am
You will not find an accurate trajectory equation for a system with air resistance, because the equations describing the drag force contain constants which can change depending on temperature, orientation of the object, ect. And it’s difficult to account for these variables, and it can easily result in chaotic behavior.
The equations for the drag force of a fluid are (air is a fluid)
F = -b v + ½ ρ v² A C
where
F is the force of drag,
ρ is the density of the fluid,[3]
v is the speed of the object relative to the fluid,
A is the reference area,
C is the drag coefficient (a dimensionless parameter)
and
b is a constant that depends on the properties of the fluid and the dimensions of the object.
Usually, you can ignore one of these two terms, depending on whether you’re at high or low speeds. At high speeds v² > v ==> ρ v² A C > b v
At low speeds, v > v² ==> b v > ρ v² A C
For a launching a cannon ball, you’ll probably work in the region of high speeds, and can just use the equation
m y″ = -½ ρ A C ( (x′)²+ (y′)² ) – mg
m x″ = -½ ρ A C ( (x′)²+ (y′)² )
The problem in solving these equations is that C, the drag coefficient, isn’t constant. If you assume it’s constant, then you can solve these coupled differential equations to get trajectory equations.
You should look at the you wikipedia link below which does give the equation for a falling object, but not an equation of an object with a non-zero component of velocity on the x-direction.
References :
http://en.wikipedia.org/wiki/Air_resistance