The ranges and heights for two projectiles projected with the same initial velocity at angles

Question:

The ranges and heights for two projectiles projected with the same initial velocity at angles $42^{\circ}$ and $48^{\circ}$ with the horizontal are $\mathrm{R}_{1}, \mathrm{R}_{2}$ and $\mathrm{H}_{1}$, $\mathrm{H}_{2}$ respectively. Choose the correct option:

1. $\mathrm{R}_{1}>\mathrm{R}_{2}$ and $\mathrm{H}_{1}=\mathrm{H}_{2}$

2. $\mathrm{R}_{1}=\mathrm{R}_{2}$ and $\mathrm{H}_{1}<\mathrm{H}_{2}$

3. $\mathrm{R}_{1}<\mathrm{R}_{2}$ and $\mathrm{H}_{1}<\mathrm{H}_{2}$

4. $\mathrm{R}_{1}=\mathrm{R}_{2}$ and $\mathrm{H}_{1}=\mathrm{H}_{2}$

Correct Option: 2,

Solution:

Range $\mathrm{R}=\frac{\mathrm{u}^{2} \sin 2 \theta}{\mathrm{g}}$ and same for $\theta$ and $90-\theta$

So same for $42^{\circ}$ and $48^{\circ}$

Maximum height $\mathrm{H}=\frac{\mathrm{u}^{2} \sin ^{2} \theta}{2 \mathrm{~g}}$

$\mathrm{H}$ is high for higher $\theta$

So $\mathrm{H}$ for $48^{\circ}$ is higher than $\mathrm{H}$ for $42^{\circ}$

Option (2)