Ratio of minimum kinetic energies of two projectiles of same mass is 4 : 1. The ratio of the maximum height attained by them is also 4 : 1. The ratio of their ranges would be :

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Ratio of minimum kinetic energies of two projectiles of same mass is 4 : 1. The ratio of the maximum height attained by them is also 4 : 1. The ratio of their ranges would be :

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16 : 1

4 : 1

8 : 1

2 : 1

answer is B.

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Detailed Solution

$\frac{{KE}_{min_{1}}}{{KE}_{min_{2}}} = \frac{\frac{1}{2} {mu}_{1}^{2} \cos^{2} θ_{1}}{\frac{1}{2} {mu}_{2}^{2} \cos^{2} θ_{2}} = \frac{4}{1} \Rightarrow \frac{u_{1} \cos θ_{1}}{u_{2} \cos θ_{2}} = \frac{2}{1} \frac{{PE}_{max_{1}}}{{PE}_{max_{2}}} = \frac{\frac{u_{1}^{2} \sin^{2} θ_{1}}{2 g}}{\frac{u_{2}^{2} \sin^{2} θ_{2}}{2 g}} = \frac{4}{1} \Rightarrow \frac{u_{1} \sin θ_{1}}{u_{2} \sin θ_{2}} = 2 \frac{R_{1}}{R_{2}} = \frac{\frac{u_{1}^{2} 2 \sin θ_{1} \cos θ_{1}}{g}}{\frac{u_{2}^{2} 2 \sin θ_{2} \cos θ_{2}}{g}} = 2 \times 2$