Find the condition that if box once comes to rest, it remains at rest (somewhere in between the top and bottom).

V_{0}^{2} = \frac{2 g s i n^{2} α}{A s i n α}

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V_{0}^{2} = \frac{3 g s i n^{2} α}{A c o s α}

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V_{0}^{2} = \frac{2 g c o s^{2} α}{A s i n α}

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V_{0}^{2} = \frac{3 g c o s^{2} α}{A s i n α}

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Solution

The correct option is B $V_{0}^{2} = \frac{3 g s i n^{2} α}{A c o s α}$
The box will come to rest if final velocity becomes zero before reaching highest point.
$\frac{m v_{0}^{2}}{2} = m g s i n α x + \int_{0}^{x} A x m g c o s α d x$
or $v_{0}^{2} = 2 g s i n α x + A g c o s α x^{2}$
it will remain at rest if the net force at that place is zero i.e.,
$μ m g c o s a = m g s i n a$
or $A x c o s a = s i n a$ or $x = \frac{s i n α}{A c o s α}$
$∴ v_{0}^{2} = 2 g s i n a (\frac{s i n α}{A c o s α}) + A g c o s a {(\frac{s i n α}{A c o s α})}^{2}$
$= \frac{3 g s i n^{2} α}{A c o s α}$

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