Q122 / 180 · KCET 2016 · Physics

Three projectiles \( A, B \) and \( C \) are projected at an angle of \( 30^{\circ}, 45^{\circ}, 60^{\circ} \) respectively. If
\( R_{A}, R_{B} \) and \( R_{C} \) are ranges of \( A, B \) and \( C \) respectively, then (velocity of projection is same for \( A \),
\( B \& C) \)

  • A
    \( R_{A}=R_{B}=R_{C} \)
  • B
    \( R_{A}=R_{C}>R_{B} \)
  • C
    \( R_{A} < R_{B} < R_{C} \)
  • Correct answer
    D
    \( R_{A}=R_{C} < R_{B} \)

Solution

Given, angle of projection of $A=30^{\circ} ;$ angle of projection of $B=45^{\circ} ;$ angle of projection of $C=60^{\circ} .$ Range of $A=R A ;$ rangeof $B=R_{B}$ irange of $C=R_{C}$ Now, Horizontal range $=\frac{(\text { Initial velocity })^{2} \times \sin (2 \times \text { Launch angle })}{2 \times \text { Acceleration due to gravity }}$ $\Rightarrow R=\frac{v_{0}^{2} \sin 2 \theta}{g}$ $\Rightarrow R \propto \sin 2 \theta$ $\therefore R_{A} \propto \sin \left(2 \times 30^{\circ}\right)=\sin \left(60^{\circ}\right)$ $\Rightarrow R_{A}=\frac{\sqrt{3}}{2}$ $R_{B} \propto \sin \left(2 \times 45^{\circ}\right)=\sin \left(90^{\circ}\right)$ $\Rightarrow R_{B}=1$ $R_{C} \propto \sin \left(2 \times 60^{\circ}\right)=\sin \left(120^{\circ}\right)$ $\Rightarrow R_{C}=\frac{\sqrt{3}}{2}$ Thus, $R_{A}=R_{C} < R_{B}$

Hence, the correct answer is option: 'D'.

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