- correct problem appears in the next post

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- Jun 13th 2009, 07:02 AMcharliebrownKinematics problem
- correct problem appears in the next post

- Jun 14th 2009, 10:19 AMthe_doc
To begin with I'll just make this question more readable and fill in the bits that have obviously been removed in the copy and paste. Also I've corrected any equations that I think haven't been written correctly. Let me know if I guess the missing bits incorrectly. I've highlighted the bits of text I added in red:

Quote:

**1.**A pilot attempts to fly with constant speed $\displaystyle v$ from the point $\displaystyle P = (D, 0)$ on the $\displaystyle x$-axis to the origin $\displaystyle O = (0; 0)$. A wind blows with speed $\displaystyle w$ in the positive $\displaystyle y$ direction. The pilot is not familiar with vector addition and thinks the shortest path to $\displaystyle O$ is achieved by flying his plane so that it always points directly towards $\displaystyle O$.

**(a)**Show that the actual flight path of the plane (in Cartesian coordinates) is given by

$\displaystyle y(x) = f(x) \sinh [g(x)]$ ; (1)

where $\displaystyle f(x)$ and $\displaystyle g(x)$ are scalar functions of $\displaystyle x$ that are to be determined.

**(b)**Consider the three cases $\displaystyle w > v$, $\displaystyle w = v$, and $\displaystyle w < v$ separately, and determine in which cases the pilot actually reaches $\displaystyle O$.

**(c)**Show that the flight path of the plane (in polar coordinates) is given by the polar equation

$\displaystyle r = r(\theta) = D ( \cos \theta )^{c_1} (1 + \sin \theta )^{c_2}$ ; (2)

where $\displaystyle c_1$ and $\displaystyle c_2$ are constants (that depend on $\displaystyle v$ and $\displaystyle w$) that are to be determined.

**(d)**Show that the radial component of the acceleration $\displaystyle a_r = 0$ and determine the transverse component $\displaystyle a$ in terms of the polar coordinates $\displaystyle r$ and $\displaystyle \theta$ (at time $\displaystyle t$).

**(e)**Determine the flight path of the plane (in Cartesian coordinates) if the wind is blowing with a velocity $\displaystyle w$ in a direction that makes an angle with the vertical (i.e., the $\displaystyle y$ axis).

- Jun 14th 2009, 02:55 PMthe_doc
It's late in the UK right now so will do so tomorrow now.

- Jun 14th 2009, 05:34 PMcalculusfrk
Hi!

I have been struggling with this problem for quite some time so would also appreciate it if you would be able to post an answer ASAP or at least some clarification!

Thanks for your assistance! - Jun 14th 2009, 06:35 PMcharliebrown
hey, yes the corrections are right. not sure why it didn't include them when i pasted the question. thanks for changing them

- Jun 15th 2009, 09:31 AMthe_doc
Here are the solutions

**(a)**to**(d)**:

**(a)**

__Spoiler__:

**(b)**

__Spoiler__:

**(c)**

__Spoiler__:

**(d)**

__Spoiler__:

Will put up solution to**(e)**in a little while (going to have my supper!).

If anyone else would like to put up the solution to**(e)**in the meantime they are welcome to! - Jun 15th 2009, 10:10 AMzorop
I love ya~~~(Clapping)!!!

- Jun 15th 2009, 11:51 AMcharliebrown
Thanks, I ended up figuring out a,b and c but your solution to d helped alot.

for e however, i'm still clueless. midterm soon but i feel much more prepared now. - Jun 16th 2009, 06:31 AMthe_docConcluding part!
**(e)**

__Spoiler__: