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# A shot fired from a gun with a speed of V at an angle $\theta$ strikes an object at a point P on the horizontal plane through the point of projection. If the object at P starts to move away from the gun with uniform acceleration a when the gun is fired, then the elevation is to be changed to ϕ in order to strike the moving object. If $\dfrac{g}{a} = \left( {\dfrac{{1 - \cos n\phi }}{{\sin n\phi - \sin n\theta }}} \right)$, find n.

Last updated date: 13th Jun 2024
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Hint: In this question, we will use the relation of the horizontal range R, time T to find the speed S of the shot. This will give us a final relation between range and speed, using this relation we will deduce the required expression and get the result. Further, we will discuss the basics of time of flight and how height changes the energy of an object.
Formula used:
$R = \dfrac{{{v^2}\sin 2\theta }}{g}$
$T = \dfrac{{2v\sin \phi }}{g}$

Here, we will use the relation of the horizontal range R, time T to find the speed of the shot, which is given as
As we know, the horizontal range R is given by:
$R = \dfrac{{{v^2}\sin 2\theta }}{g}$
Time of flight T is given by:
$T = \dfrac{{2v\sin \phi }}{g}$
So, the distance moved by P in the given time T is given by:
$S = \dfrac{1}{2}a{\left( {\dfrac{{2v\sin \phi }}{g}} \right)^2}$
Therefore, the changed range is given by the addition of horizontal range R and distance moved by P in given time T, this can be written as:
$\dfrac{{{v^2}\sin 2\phi }}{g} = \dfrac{{{v^2}\sin 2\theta }}{g} + \dfrac{{2a{v^2}{{\sin }^2}\phi }}{{{g^2}}}$
$\Rightarrow g\sin 2\phi = g\sin 2\theta + 2a{\sin ^2}\phi$
Now, we solve for R.H.S:
$g\sin \theta + 2a\left( {\dfrac{{1 - \cos 2\phi }}{2}} \right)$
$= g\sin 2\theta + a - a\cos 2\phi$
Now, again when we put this value of R.H.S in above equation we get:
$\Rightarrow g\sin 2\phi - g\sin 2\theta = a - a\cos 2\phi$
Solving for n, we get:
$\Rightarrow g(\sin 2\phi - \sin 2\theta ) = a(1 - \cos 2\phi )$
\eqalign{& \Rightarrow \dfrac{g}{a} = \dfrac{{1 - \cos 2\phi }}{{\sin 2\phi - \sin 2\theta }} \cr & \therefore n = 2 \cr}
Therefore, we get the required result, i.e., n=2, which is given by the above value.

Here, we should remember that time of flight can be calculated by wave as well. Also, total energy is given by the sum of potential energy and kinetic energy. The unit of energy here is electron volt which is $1.602 \times {10^{ - 19}}Joule$ . The S.I unit of energy is Joule. At the maximum height the potential energy is maximum and kinetic energy is zero