Solve Time to Speed with Drag Equation

[Diresu]

(this is for a hobby, not homework)

I fully understand the drag equation and can work it forwards and backwards.
http://en.wikipedia.org/wiki/Drag_equation

How can I use the same data (plus mass if needed) to determine time to top speed or a given speed?

I'm sure there is a simple formula related to the Drag Equation that gives time to speed. I am looking everywhere and I can't find what I need.

Thanks a billion in advance for a shove in the right direction!

 

[xez]

I don't think you can find the equation you seek just
because that's not all the information that's required.

The drag equation gives the FORCE of the drag given
velocity, et. al. but that says nothing about the
FORCE or power or accelleration your vechicle can
produce given a certain present velocity and drag forces
acting upon it.

You could say you have a fixed force of propulsive power
available and a fixed mass of vehicle then in that case
the net force on the vehicle would be
F_net = F_propulsive - F_drag, and then via F = m*a
your accelleration at any moment would be F_net/m, and
the integral of accelleration(time)=F_net(time)/mass
would be velocity(time), and that'd be a curve that'd
even out and stay level when F_net=0 due to top speed
having been reached.

If you had a vehicle that was capable of a certain constant
power output you could similarly calculate a resultant
accelleration due to the force balance and again get
to a velocity and top speed.

More realistically, though, the power available from
most engines will depend on their RPM, and consequently
the gear ratio and consequently on the present velocity,
so you have several complications to determine the
actual accelleration.

 

[chanvincent]

I'm sure there is a simple formula related to the Drag Equation that gives time to speed.

xez was right...

Unless you made a simple assumtion, i.e. the force provided by the vihicle is constant, the answer is anything but far from simple...

 

[Diresu]

Thanks xez and chanvincent, you guys rock!

No easy answer? Dang!

Ok maybe if I was more specific this would make more sense.

I am talking about a theoretical aircraft with constant thrust.

From the drag equation you can determine top speed

Velocity(top speed)=Sqrt[(2T)/(P*A*Cd)]

Here is a simple example
T(thrust of engine) = 10000 Newtons
P(atmospheric density) = 1.25 Kg/M^3
A(reference area of craft) = 10 square meters
Cd(drag coefficient) = .1

The answer is 455.4 Km/h (have to change from M to Km)

Isn't there a companion equation that determines time to a given speed or top speed?

My biggest problem is that acceleration doesn't operate in a straight line. It starts out accelerating rapidly and gradually slows to 0 acceleration. Is there no simple time to speed equation for simple models lke this? I believe it, I just can't believe it!

 

[Diresu]

Whooops...

Hypothetically speaking the craft weighs 1000 Kg (if that is the right kind of units).

If there is no obvious answer that is fine. If there is an available equation that is a decent enough test number.

 

[chanvincent]

Well, as I said be4, the answer is not simple if the force is not constant.

I am talking about a theoretical aircraft with constant thrust.

But now you are talking about the constant force.. then the solution IS very simple,

You have the drag force
$$F_d = -\frac{1}{2}\rho v^2 C_p A$$
and the force from the trust
$$F_{thrust} = \mbox{constant}$$,

now the total force

$$\Sigma F = F_d + F_{trust} = Ma$$
$$-\frac{1}{2}\rho v^2 C_p A+ F_{trust} = Ma$$
$$-\frac{1}{2}\rho v^2 C_p A+ F_{trust} = M\frac{dv}{dt}$$

I will leave this first order non-linear DE here for you to work with...
Notice, luckily, there is an exact solution for this DE...

 

[Diresu]

chanvincent

YOU ROCK!

This might sounds like a retarded question, but... I am not an expert.

I don't totally understand the first line.
Total Force = Force of drag + Force of thrust? = Mass? (Ma = Mass?)

Second line I get because it is the drag equation + Force of thrust? = Mass?

The third line what is M(dv/dt)?

DOH!

Thanks a billion!

 

[chanvincent]

a is accelaration... dv/dt is also accelaration...
You havn't learned calculus yet, have you?
otherwise you should be familiar with dv/dt...
well... Since you claimed that you fully understand the drag equation, I thought you knew calculus...

although you do not know how do derive the answer... maybe you want to take a look...
http://en.wikipedia.org/wiki/Drag_(physics)

Hopefully you could understand... Cheer...

 

[Diresu]

Thanks Chan

Actually I understand basic calculus and derivatives. 3X^2 = 6x and all that stuff.

dv/dt being acceleration makes sense, velocity over time.

But what the heck is M in Ma? Mass, weight, something else completely?

THANKS!

 

[chanvincent]

m = mass, a = accelration

F=ma, => Total Force equals to mass times accelration...

understand?

 

[Diresu]

Ok, awesome, I didn't spot the F=ma in that equation.

So now that I have velocity over time on one side I can solve for time? That I can certainly do.

One last tiny question on something I always screw up... my units.

If I'm using Newtons for thrust, I should use Kilograms for Mass? Velocity (dv) is obviously meters per second. Time (dt) obviously seconds.

 

[chanvincent]

If I'm using Newtons for thrust, I should use Kilograms for Mass? Velocity (dv) is obviously meters per second. Time (dt) obviously seconds.

Yes...

So now that I have velocity over time on one side I can solve for time? That I can certainly do.

It is not as easy as you think... I doubt that you could really solve it... but, yes, it could be solved...

 

[Diresu]

So

dt=(M*dv)/(Fd+Fthrust)

Did I solve that right?

 

[chanvincent]

Yes... you got it... the answer should look like some hyperbolic tangent... just go ahead...

 

[Diresu]

YOU ROCK!

Ok, I got to go to bed but I'm going to start plugging in practice data tomorrow afternoon.

Thank you so much, I could have never EVER done it without your help. It doesn't look too bad once it is finished but I was clueless.

 

[Diresu]

Chanvincent

dt=(M*dv)/(Fd+Fthrust)

Fd+Fthrust = 0 at top speed. That makes sense because once the force of Thurst and Drag even out there is no longer any acceleration. But in F=Ma if F = 0 then a = 0 too.

dt=(M*dv)/(Fd+Fthrust) = zero no matter what data is plugged in.

I must be 1 nanometer away from the correct solution but... what the heck?

I do understand the basic concept of derivitives meaning, 4X^2 = 8x, but I honestly don't know when or why they are used. I'm beggin' you here, I must be close but what do I do with it? I'm going to use this thing a million times once I am done but... Dag!

 

[chanvincent]

I told you the integral is not easy to do (in your level)

$$dt = \frac{Mdv}{F_{trust}-\frac{1}{2}\rho v^2 C_p A}$$
$$\int_0^Tdt=\int_0^v \frac{Mdv}{F_{trust}-\frac{1}{2}\rho v^2 C_p A }$$

Any idea how do solve this? Hint:

$$\frac{d}{dx} tanh^{-1}x = \frac{1}{1 - x^2}$$

 

[Diresu]

Thanks chan!

Sadly I've been sworn off working on this thing until next week.

That integral does look like a monster. By if there is an ordered set of steps I'm sure I can do it. I'm going to save that and come back to it next week.

Thanks!