- #1
v6maik
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Hello,
I'm working on a project about solid fuel rockets and since these are moving through the atmosphere, they experience Air-drag. I'm trying to set up a formula to exactly calculate the height a rocket will achieve. So without the use of any model. The problem I ran into is that I can't get the formula for acceleration to include air-drag, since I ran into the following loop:
-the speed of the rocket (and thereby the acceleration) depends on the air-drag.
-the air-drag depends on the speed of the rocket (and thereby the acceleration)
I found this equation for air-drag but it's of no use in its current form.
Fd(t)= -0,5 * p * A * Cd * v^2
as you can see, drag depends on the speed relative to the air-mass, which is pretty obvious.
But since v=a * t, this formula is the same as:
Fd(t)= -0,5 * p * A * Cd * (a*t)^2
and since a= F/m this formula is the same as:
Fd(t)= -0,5 * p * A * Cd (((F / m ) *t)^2
note that F is the net Force on the rocket. The net Force at a given time is equivalent to the propulsion force minus gravity minus drag:
Fnet(t)= Fp(t) - Fg(t) - Fd(t)
So the drag formula now is:
Fd(t)= -0,5 * p * A * Cd (((Fm(t) - Fg(t) - Fd(t)) / m ) *t)^2
Notice that this formula Fd(t) involves it's own answer, so it is a differential equasion, right?
Now, I can simplify this formula to this, leaving 3 constants: a, b and c:
Fd(t)= a * ( (b-Fd(t) )/c * t)^2
Any suggestions about solving this problem? Or might there be a different equation to calculate air-drag at a given time during acceleration?
Thanks ahead!
Kind regards,
Maik
I'm working on a project about solid fuel rockets and since these are moving through the atmosphere, they experience Air-drag. I'm trying to set up a formula to exactly calculate the height a rocket will achieve. So without the use of any model. The problem I ran into is that I can't get the formula for acceleration to include air-drag, since I ran into the following loop:
-the speed of the rocket (and thereby the acceleration) depends on the air-drag.
-the air-drag depends on the speed of the rocket (and thereby the acceleration)
I found this equation for air-drag but it's of no use in its current form.
Fd(t)= -0,5 * p * A * Cd * v^2
as you can see, drag depends on the speed relative to the air-mass, which is pretty obvious.
But since v=a * t, this formula is the same as:
Fd(t)= -0,5 * p * A * Cd * (a*t)^2
and since a= F/m this formula is the same as:
Fd(t)= -0,5 * p * A * Cd (((F / m ) *t)^2
note that F is the net Force on the rocket. The net Force at a given time is equivalent to the propulsion force minus gravity minus drag:
Fnet(t)= Fp(t) - Fg(t) - Fd(t)
So the drag formula now is:
Fd(t)= -0,5 * p * A * Cd (((Fm(t) - Fg(t) - Fd(t)) / m ) *t)^2
Notice that this formula Fd(t) involves it's own answer, so it is a differential equasion, right?
Now, I can simplify this formula to this, leaving 3 constants: a, b and c:
Fd(t)= a * ( (b-Fd(t) )/c * t)^2
Any suggestions about solving this problem? Or might there be a different equation to calculate air-drag at a given time during acceleration?
Thanks ahead!
Kind regards,
Maik