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Angle between a Plane and a Line

A projectile ...

Question

A projectile is fired at an angle $θ$ with the horizontal. Find the condition under which it lands perpendicular on an inclined plane of inclination $α$ as shown in figure.

s i n α = c o s (θ - α)

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c o s α = s i n (θ - α)

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t a n θ = c o t (θ - α)

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c o t (θ - α) = 2 t a n α

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Solution

The correct option is D $c o t (θ - α) = 2 t a n α$

$a_{x} = - g s i n α a_{y} = - g c o s α$
$u_{x} = u c o s (θ - α)$
$u_{y} = u s i n (θ - α)$

Time of flight:
$y = u_{y} t + \frac{1}{2} a_{y} t^{2}$
$0 = u s i n (θ - α) t - \frac{1}{2} g (c o s α) t^{2} t = \frac{2 u s i n (θ - α)}{g c o s α}$

At this time, $v_{x} = 0$ for perpendicular landing

$v_{x} = u_{x} + a_{x} t$
$0 = u c o s (θ - α) - g s i n α . \frac{2 u s i n (θ - α)}{g c o s α}$
$2 t a n α = c o t (θ - α)$

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A projectile is fired at an angle θ with the horizontal. Find the condition under which it lands perpendicular on an inclined plane of inclination α as shown in figure.

A projectile is fired at an angle $θ$ with the horizontal. Find the condition under which it lands perpendicular on an inclined plane of inclination $α$ as shown in figure.