Kinematics

JEE Main level, single correct, medium

260 of 308

Question 260 of 308

Two particles are projected from the same point with the same speed u such that they have the same range R, but different maximum heights, h1{h}_{1} and h2.{h}_{2}. Which of the following is correct?

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Answer: (D) R2=16h1h2{R}^{2}=16{h}_{1}{h}_{2}

We get same range for angle of projections θ\theta and 90−θ90-\theta .
h=u2sin2θ2gh=\dfrac{{u}^{2}{sin}^{2}\theta }{2g}
⇒h1=u2sin2θ2g,h2=u2cos2θ2g\Rightarrow {h}_{1}=\dfrac{{u}^{2}{sin}^{2}\theta }{2g},{h}_{2}=\dfrac{{u}^{2}{cos}^{2}\theta }{2g}
h1h2=u4sin2θcos2θ4g2=4u4sin2θcos2θ16g2{h}_{1}{h}_{2}=\dfrac{{u}^{4}{sin}^{2}{\theta cos}^{2}\theta }{4{g}^{2}}=\dfrac{{4u}^{4}{sin}^{2}\theta {cos}^{2}\theta }{16{g}^{2}}
=(u2sin2θg)2116=R216={\left(\dfrac{{u}^{2}sin2\theta }{g}\right)}^{2}\dfrac{1}{16}=\dfrac{{R}^{2}}{16}
∴R2=16h1h2∴{R}^{2}=16{h}_{1}{h}_{2}

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