So the problem I'm trying to solve is prefaced with this:
Earlier we mentioned that NASA claims that the Vomit Comet can make passengers experience weightlessness for about 25 seconds. Let’s check on that claim. To simulate weightlessness (neutral buoyancy) the pilot must execute a parabolic flight path: $y = Ax^2 + Bx + C.$ In Problem #88 you should have found that B and C were 1 and 7000, respectively, so the flight path is $y = Ax^2 + x + 7000$ with A yet to be determined. The pilot will climb at an angle of 45◦ to an altitude of about 7000 meters and then follow this parabolic path to produce a vertical acceleration of d $d^2y/dt^2 = −9.8 m/s^2$ (matching the acceleration due to gravity) and horizontal acceleration of d 2x dt 2 = 0. This will provide neutral buoyancy inside the plane. On the way back down the pilot pulls out of this dive when the altitude returns to 7000 meters. For training purposes this is repeated 40 times.
And this is the problem itself:
To determine A we need one more fact. At the beginning of the maneuver, the initial airspeed is about 180 meters/second (approximately 400 mph). Use this to determine dx/dt and in turn use this and the fact that $d^2y/dt^2$ = −9.8 to determine A.
The solution is $A = -9.8 / 180^2$.
I think I am confused about some of the concepts involved in solving this problem, which are detailed here along with things I know:
I'm not sure how to achieve the "speed" of things in calculus. I am used to velocity, which you find by taking the derivative of whatever you're working with, and acceleration is found with the second derivative. So, how is speed calculated?
I understand that airspeed is the sum of windspeed and groundspeed. In order to visualize this, I drew a right triangle with the airspeed being the diagonal (which in this case I believe is curved, so I guess it's technically not a right triangle, but for simplicity's sake this is what I used), the bottom being the groundspeed ($dx/dt$) , and the side being the windspeed ($dy/dt$). I am guessing in order to solve this I would need to utilize Pythagorean's theorem, $a^2+b^2=c^2$, which in this problem is $180^2 = a^2 + b^2$. What I don't understand about this is how is this relevant to finding A?
From what I understand, A is supposed to represent how wide the parabola is.
This is what I did to determine $dx/dt$:
$dy = 2Axdx + dx$
$dx = dy/2Ax+1$
$dx/dt = 1/(2Ax+1)$
I don't understand how I would use this and the fact that $d^2y/dt^2 = −9.8 m/s^2$ to solve for A.