A man is sitting on the shore of a river. He is in the line of a 1.0 m long boat and is 5.5 m away from the center of the boat. He wishes to throw an apple into the boat. If he can throw the apple only with a speed of 10 m/s, find the minimum and maximum angles of projection for successful shot. Assume that the point of Projection and the edge of the boat are in the same horizontal level.

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Solution

The correct option is
A and B

Let AB be the boat, to touch A, Range should be 5m.
u= 10 $\frac{m}{s}$ , g = 10m/ $s^{2}$ , R = 5 m,
$∴$ R = $\frac{u^{2} s i n 2 θ}{g}$ $\Rightarrow$ 5 = $\frac{10^{2} s i n 2 θ}{10}$ $\Rightarrow$ sin 2 $θ$ = $\frac{1}{2}$
$∴$ 2 $θ$ = $30^{\circ}$ ,15 $θ^{\circ}$ (as in $π$ - $θ$ =sin $θ$ )
$∴$ $θ$ = $15^{\circ}$ , $75^{\circ}$
To throw at B
R = 6m
$∴$ 6 = $\frac{10^{2} s i n 2 θ}{10}$ $\Rightarrow$ sin 2 $θ$ = $\frac{3}{5}$
$\Rightarrow$ 2 $θ$ = $37^{\circ}$ , $143^{\circ}$ (sin( $π$ - $θ$ )= sin $θ$ )
$\Rightarrow$ $θ$ = ${18.5}^{\circ}$ , ${71.5}^{\circ}$
$∴$ Minimum is $15^{\circ}$ and maximum is $75^{\circ}$
$∴$ For a successful shot,
$15^{\circ}$ $\leq$ $θ$ $\leq$ ${18.5}^{\circ}$ and ${71.5}^{\circ}$ $\leq$ $θ$ $\leq$ $75^{\circ}$
$∴$ Minimum is $15^{\circ}$ and maximum is $75^{\circ}$ . .

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