The original question was:

A fighter plane is flying horizontally at a height of 250 m from ground with constant velocity of 500m/s.It passes exacily over a cannon which can fire a shell at any time in any direction with a speed of 100 m/s. Find the duration of time for which the plane is in danger of being hit by a cannon shell.

Note: Just to clear any confusions, the plane is passing exactly over the canon, so the whole action is taking place in 2D.

My attempt:

The equation of trajectory for the cannon shell is: $$y=x\tan\theta -\frac{gx^2\sec^2\theta}{2u^2}$$

Now comes the part where I will have to find the boundary till where the danger zone for the plane extends to.

The questions that came up at this stage were:

1) With respect to which variable($x,y,\theta)$ should the equation of trajectory should be partially differentiated?

2) What would be the significance or what would the equations represent physically after it has been differentiated with any of the three variables $(x,y,\theta)$?

3) In order to obtain the maximum range of danger zone, which variable(s) must be maximized?

However, the answers for Q.1 & Q.2 aren't yet completely and convincingly known to me but my opinion on Q.3 is as follows:

For maximum range of the danger zone, I will need to adjust my canon in such a way that it adjusts its $\theta$(projection angle) to give maximum x & y.

Therefore, differentiating the $x$ or $y$ with respect to $\theta$ and equating $\frac{\partial y}{\partial\theta}=0$,

We obtain, $\tan\theta=\frac{u^2}{gx}$ . Substituting this value of $\tan\theta$ we can obtain the maximum area of danger zone.

I would like to know the answers to my first (if my answer to the 3rd question is correct then the answer for the first question is clear) and second question and whether my opinion on the third question is correct or not.

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    $\begingroup$ Please note that we don't answer homework or worked example type questions. Please see this page in the site help for more on what topics you can ask about here. $\endgroup$ – John Rennie Apr 16 '19 at 5:49
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    $\begingroup$ @JohnRennie what if I just remove my original question...then I guess it wouldn't like a homework question...the main objective of the question is to know the significance of the differentiation of equation of trajectory wrt to different variables...the example was just provided to give a background and my question some relevance...I don't see it as a homework question $\endgroup$ – Carrick Apr 16 '19 at 5:58
  • $\begingroup$ Hi Carrick. Welcome to Phys.SE. If you haven't already done so, please take a minute to read the definition of when to use the homework-and-exercises tag, and the Phys.SE policy for homework-like problems. $\endgroup$ – Qmechanic Apr 16 '19 at 7:03
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    $\begingroup$ @Qmechanic I just read it sir...and I would like to make it clear that I am not interested in knowing the correct method(I've pretty much figured it out myself) to that example, rather I'm very curious about knowing what's written in Q2 above. That's the main objective of my question. $\endgroup$ – Carrick Apr 16 '19 at 7:09
  • $\begingroup$ Sad...but I guess physics.SE is turning out to be a major waste of time for me $\endgroup$ – Carrick Apr 30 '19 at 4:50