The ranges and heights for two projectiles projected with the same initial velocity at angles 42$^\circ$ and 48$^\circ$ with the horizontal are R₁, R₂ and H₁, H₂ respectively. Choose the correct option :

Here, two projectiles are projected at angles 42$^\circ$ and 48$^\circ$ with same initial velocity.<br/><br/>As we know the expression of range of projectile, <br/><br/>Range $= {{{u^2}\sin 2\theta } \over g}$<br/><br/>AT $\theta$₁ = 42$^\circ$,<br/><br/>Range, ${R_1} = {{{u^2}\sin 2(42)^\circ } \over g} = {{0.99{u^2}} \over g}$<br/><br/>At $\theta$₂ = 48$^\circ$ <br/><br/>Range, ${R_2} = {{{u^2}\sin 2(48)^\circ } \over g} = {{0.99{u^2}} \over g}$<br/><br/>The range of the projectile is same for the two projectiles.<br/><br/>Therefore, R₁ = R₂<br/><br/>Now, as we know the expression of height of the projectile,<br/><br/>${H_{\max }} = {{{u^2}\sin \theta } \over {2g}}$<br/><br/>At 42$^\circ$, $${H_{\max }} = {H_1} = {{{u^2}\sin 42^\circ } \over {2g}} = {{0.669{u^2}} \over {2g}}$$<br/><br/>At 48$^\circ$, $${H_{\max }} = {H_2} = {{{u^2}\sin 48^\circ } \over {2g}} = {{0.743{u^2}} \over {2g}}$$<br/><br/>Higher the value of $\theta$ higher the value of maximum height. Therefore, H₁ < H₂.