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Q.
Two projectiles of same mass have their maximum kinetic energies in ratio $4 : 1$ and ratio of their maximum heights is also $4 : 1$ then what is the ratio of their ranges ?
The kinetic energy of the projectiles is maximum at the point of release. The ratio of K.E. of projectiles
$\frac{\frac{1}{2}mu_{1}^{2}}{\frac{1}{2}mu_{2}^{2}}=\frac{4}{1}$
$\frac{u^{2}_{1}}{u^{2}_{2}}=\frac{4}{1}\quad..... \left(i\right)$
The ratio of maximum heights of projectiles
$\frac{H_{1}}{H_{2}}=\frac{4}{1}$
$\frac{\frac{u_{1}^{2} sin^{2} \theta _{1}}{2g}}{\frac{u_{2}^{2} sin^{2} \theta _{2}}{2g}}=\frac{4}{1}$
$\frac{u_{1}^{2}}{u_{2}^{2}}\times\frac{sin^{2} \theta_{1}}{sin^{2} \theta_{2}}=\frac{4}{1}$
$\frac{4}{1}\times\frac{sin^{2} \theta_{1}}{sin^{2} \theta_{2}}=\frac{4}{1}\quad$ (from equation $\left(i\right)$)
$sin^{2} \theta_{1} = sin^{2} \theta_{2}$
$\theta_{1}=\theta_{2}$
Now, ratio of ranges of projectiles
$\frac{R_{1}}{R_{2}}=\frac{u_{1}^{2} sin 2 \theta_{1}}{u^{2}_{2} sin 2 \theta_{2}}$
$\frac{R_{1}}{R_{2}}=\frac{u_{1}^{2} sin 2 \theta _{1}}{u^{2}_{2} sin 2 \theta _{1}}\left( \therefore \theta_{1}=\theta_{2}\right)$
$\frac{R_{1}}{R_{2}}=\frac{u_{1}^{2} }{u^{2}_{2} }, \frac{R_{1}}{R_{2}}=\frac{4}{1}$