Rate of change problem my teachers can't solve

Flyingismylife

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I'm currently doing a math exploration in my math higher level class on air navigation. After handing in my first draft of my essay, my teacher pointed out that one of my examples would make a very nice rates of change question. But as it turns out, it's not very nice at all.

Here's a link to the problem: https://www.dropbox.com/s/uxs6nfcuso58n88/Essay problem.pdf

I want to find an equation which solves for d∆h/dt, but the answer I get contains another rate which is unknown (dVa/dt).

Please help. I've been stuck on this forever, and my teachers can't seem to be able to solve it themselves.

Best regards,
Filip
UWC Red Cross Nordic
 
I'm currently doing a math exploration in my math higher level class on air navigation. After handing in my first draft of my essay, my teacher pointed out that one of my examples would make a very nice rates of change question. But as it turns out, it's not very nice at all.

Here's a link to the problem: https://www.dropbox.com/s/uxs6nfcuso58n88/Essay problem.pdf

I want to find an equation which solves for d∆h/dt, but the answer I get contains another rate which is unknown (dVa/dt).

Please help. I've been stuck on this forever, and my teachers can't seem to be able to solve it themselves.

Best regards,
Filip
UWC Red Cross Nordic
First thing I notice when reading the statement of the problem is that since wind velocity is a vector, a change is also an a vector. Are you assuming the wind does not change direction?

The units of the rate k (and its components) are acceleration. You can make time (t) the independent variable and take derivatives of v (i.e., of the velocity components) with respect to t.

Please show us what derivatives you have tried to take.
 
Hi,

Nothing seems to appear when I try to reply to the thread. It might just be that the site needs time to update, but just to be safe I'm sending you this personal message.

Yes, I do assume that the direction of wind does not change, and wind is indeed a velocity vector with the acceleration k.

Here's how I tried to solve the problem, which, as shown in my solution, results in the unknown rate of the true airspeed (dVa/dt): https://www.dropbox.com/s/k4bccr9g3i...20solution.pdf

Thank you for your early reply,

//Filip
You are mixing position vectors and velocity vectors! Your construction has to be that the vector sum of va\displaystyle \vec{v_a} and vw\displaystyle \vec{v_w} must be in the direction of the destination. But to say the ground speed is equal to the distance AB is nonsense, since they don't have the same units.

I would project the two velocities into components along AB and perpendicular to AB. The transverse components must cancel, from which you find angle Δh\displaystyle \Delta h. You know angles w and d, but not h.

vw sin(wd)=va sin(Δh)\displaystyle \displaystyle |v_w|\ \sin(w - d) = |v_a|\ \sin(\Delta h)

Since airspeed and angles w, d do not change, an increase of vw\displaystyle |v_w| requires a proportional increase of sin(Δh)\displaystyle \sin(\Delta h).

Knowing Δh\displaystyle \Delta h, the ground speed is

vg=va cos(Δh)+vw cos(wd)\displaystyle \displaystyle v_g = |v_a|\ \cos(\Delta h) + |v_w|\ \cos(w - d)

EDIT: This appears to be what you did, except for simplifying before differentiating. The magnitude of the airspeed is almost surely a constant of the aircraft.
 
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Thank you for clearing things up. Really helpful!

However, if we were to say that the airspeed is not a constant of the aircraft, and that the pilot will adjust the airspeed in order to keep the ground speed constant (unrealistic, but I'm more interested in the math of it all), how do you find d∆h/dt without ending up with the unknown and problematic rate of change in airspeed dVa/dt?

Again, thanks a million for you help!
 
Thank you for clearing things up. Really helpful!

However, if we were to say that the airspeed is not a constant of the aircraft, and that the pilot will adjust the airspeed in order to keep the ground speed constant (unrealistic, but I'm more interested in the math of it all), how do you find d∆h/dt without ending up with the unknown and problematic rate of change in airspeed dVa/dt?

Again, thanks a million for you help!
If you keep vector vg\displaystyle \vec{v_g} constant, then va=vgvw\displaystyle \vec{v_a} = \vec{v_g} - \vec{v_w}.
 
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