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BITSAT 2012
Exam
BITSAT
Year
2012
Subject
Physics
Chapter
Motion in Two Dimensions

BITSAT 2012 · Physics · Motion in Two Dimensions

A projectile can have the same range R for two angles of...

A projectile can have the same range R\mathrm{R} for two angles of projection. If t1t_{1} and t2t_{2} be the times of flight in two cases, then what is the product of two times of flight?

Options

  1. At1t2∝R\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{R}
  2. Bt1t2∝R2\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{R}^{2}
  3. Ct1t2∝l/R\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{l} / \mathrm{R}
  4. Dt1t2∝1/R2\mathrm{t}_{1} \mathrm{t}_{2} \propto 1 / \mathrm{R}^{2}

Correct answer

A. t1t2∝R\mathrm{t}_{1} \mathrm{t}_{2} \propto \mathrm{R}

Step-by-step solution

t1=2usin⁡θg\mathrm{t}_{1}=\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}} and t2=2usin⁡(90−θ)g=2ucos⁡θg\mathrm{t}_{2}=\frac{2 \mathrm{u} \sin (90-\theta)}{\mathrm{g}}=\frac{2 \mathrm{u} \cos \theta}{\mathrm{g}}∴t1t2=4\mathrmu2cos⁡θsin⁡θ\mathrmg2=2g[\mathrmu2sin⁡2θg]=2gR,\begin{aligned} \therefore \mathrm{t}_{1} \mathrm{t}_{2} &=\frac{4 \mathrmu^{2} \cos \theta \sin \theta}{\mathrmg^{2}}=\frac{2}{\mathrm{ g}}\left[\frac{\mathrmu^{2} \sin 2 \theta}{\mathrm{g}}\right] \\ &=\frac{2}{\mathrm{ g}} \mathrm{R}, \end{aligned}$wherewhere\mathrm{R}istherange.Henceis the range. Hencet_{1} t_{2} \propto R$
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