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प्रश्न
Water level is maintained in a cylindrical vessel up to a fixed height H. The vessel is kept on a horizontal plane. At what height above the bottom should a hole be made in the vessel so that the water stream coming out of the hole strikes the horizontal plane at the greatest distance from the vessel.
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उत्तर
It is given that H is the height of the cylindrical vessel.
Now, let h be the height of the hole from the surface of the tank.
The velocity of water \[\left( v \right)\] is given by
\[v = \sqrt{2g(H - h)}\]
Also, let t be the time of flight.
Now,
\[t = \sqrt{\frac{2h}{g}}\]
Let x be the maximum horizontal distance .
\[ \therefore x = v \times t\]
\[ = \sqrt{2g(H - h)} \times \sqrt{\frac{2h}{g}}\]
\[ = \sqrt{4(Hh - h^2 )}\]
For x to be maximum,
\[\frac{d}{dh}\left( Hh - h^2 \right) = 0\]
\[ \Rightarrow 0 = H - 2h\]
\[ \Rightarrow h = \frac{H}{2}\]
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