Question

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.

• A. ${15}^{\circ }$,${75}^{\circ }$
• B. ${18.5}^{\circ }$,${75}^{\circ }$
• C. ${15}^{\circ }$,${71.5}^{\circ }$
• D. ${18.5}^{\circ }$,${71.5}^{\circ }$

Answer 1

The correct option is A

${15}^{\circ }$,${75}^{\circ }$

Let AB be the boat, to touch A, Range should be 5m.
u= 10 $\frac{m}{s}$, g = 10m/$s^2$, R = 5 m,
$\therefore$R = $\frac{u^{2}sin2\theta }{g}$ $\Rightarrow$ 5 = $\frac{{10}^{2}sin2\theta }{10}$$\Rightarrow$ sin 2$\theta$ = $\frac{1}{2}$
$\therefore$2$\theta$ = ${30}^{\circ }$,15${\theta }^{\circ }$(as in$\pi$-$\theta$=sin$\theta$)
$\therefore$$\theta$ = ${15}^{\circ }$,${75}^{\circ }$
To throw at B
R = 6m
$\therefore$ 6 = $\frac{{10}^{2}sin2\theta }{10}$$\Rightarrow$sin 2$\theta$ = $\frac{3}{5}$
$\Rightarrow$2$\theta$ = ${37}^{\circ }$,${143}^{\circ }$(sin($\pi$-$\theta$)= sin $\theta$)
$\Rightarrow$$\theta$ = ${18.5}^{\circ }$,${71.5}^{\circ }$
$\therefore$Minimum is ${15}^{\circ }$ and maximum is ${75}^{\circ }$
$\therefore$For a successful shot,
${15}^{\circ }$$\le$$\theta$$\le$${18.5}^{\circ }$ and ${71.5}^{\circ }$$\le$$\theta$$\le$${75}^{\circ }$
$\therefore$Minimum is ${15}^{\circ }$ and maximum is ${75}^{\circ }$. .