First slide

Projectile motion

Question

A stone is to be projected horizontally from top of a 1.7 m high pole. Calculate initial velocity of projection (in ms^-1), if it strikes perpendicularly an inclined plane as shown in the figure.

Difficult

Solution

$Here, u_{x} = u, u_{y} = 0, a_{y} = - g = - 10 {ms}^{- 2}, a_{x} = 0$
$From figure, u_{x} = v_{0} \cos 53^{\circ}$
$and v_{y} = - v_{0} \sin 53^{\circ}$
$\begin{array}{l} ∵ v_{x} = u_{x} + a_{x} t \\ \Rightarrow v_{y} = u_{y} + a_{y} t = 0 - 10 t \\ Or - v_{0} \sin 53^{\circ} = - 10 t \\ ∴ v_{0} \sin 45^{\circ} = 10 t \\ \Rightarrow \frac{v_{0} \cos 53^{\circ}}{v_{0} \sin 53^{\circ}} = \frac{u}{10 t} \end{array}$
$Here, A B = \frac{1}{2} g t^{2} = 5 t^{2} and O B = u t \cot 53^{\circ}$
$\begin{array}{l} ∴ 1.7 = A B + O B = 5 t^{2} + u t \times \frac{3}{4} \\ \Rightarrow 1.7 = 5 {(\frac{4 u}{30})}^{2} + \frac{3}{4} u \times \frac{4 u}{30} \\ ∴ u^{2} = 9 \\ ∴ u = 3 {ms}^{- 1} \end{array}$

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