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3-2.Motion in Plane
medium
| Column $-I$ Angle of projection |
Column $-II$ |
| $A.$ $\theta \, = \,{45^o}$ | $1.$ $\frac{{{K_h}}}{{{K_i}}} = \frac{1}{4}$ |
| $B.$ $\theta \, = \,{60^o}$ | $2.$ $\frac{{g{T^2}}}{R} = 8$ |
| $C.$ $\theta \, = \,{30^o}$ | $3.$ $\frac{R}{H} = 4\sqrt 3 $ |
| $D.$ $\theta \, = \,{\tan ^{ - 1}}\,4$ | $4.$ $\frac{R}{H} = 4$ |
$K_h :$ kinetic energy at the highest point
A$A-1,\,\,B-2,\,\,C-3,\,\,D-4$
B$A-4,\,\,B-3,\,\,C-2,\,\,D-1$
C$A-4,\,\,B-1,\,\,C-3,\,\,D-2$
D$A-3,\,\,B-2,\,\,C-4,\,\,D-1$
Solution
$1.$ $\frac{\mathrm{K}_{\mathrm{h}}}{\mathrm{K}_{\mathrm{i}}}=\frac{\frac{1}{2} \mathrm{m}(\mathrm{u} \cos \theta)^{2}}{\frac{1}{2} \mathrm{mu}^{2}}=\cos ^{2} \theta=\frac{1}{4} \Rightarrow \theta=60^{\circ}$
$2.$ $\frac{\mathrm{gT}^{2}}{\mathrm{R}}=\frac{\mathrm{g}\left(\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}}\right)^{2}}{\frac{\mathrm{u}^{2}}{\mathrm{g}} 2 \sin \theta \cos \theta}=2 \tan \theta=8$
$\Rightarrow \tan \theta=4 \Rightarrow \theta=\tan ^{-1}(4)$
$\frac{\mathrm{R}}{\mathrm{H}}=\frac{24^{2} \sin \theta \cos \theta}{\mathrm{g}} \times \frac{2 \mathrm{g}}{4^{2} \sin ^{2} \theta}=4 \cot \theta$
$ycot$ $\theta=y \sqrt{3}$
$\cot \theta=\sqrt{3}$
$\theta=30^{\circ}$
$ycot$ $\theta=y$
$\cot \theta=1$
$\theta=45^{\circ}$
$2.$ $\frac{\mathrm{gT}^{2}}{\mathrm{R}}=\frac{\mathrm{g}\left(\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}}\right)^{2}}{\frac{\mathrm{u}^{2}}{\mathrm{g}} 2 \sin \theta \cos \theta}=2 \tan \theta=8$
$\Rightarrow \tan \theta=4 \Rightarrow \theta=\tan ^{-1}(4)$
$\frac{\mathrm{R}}{\mathrm{H}}=\frac{24^{2} \sin \theta \cos \theta}{\mathrm{g}} \times \frac{2 \mathrm{g}}{4^{2} \sin ^{2} \theta}=4 \cot \theta$
$ycot$ $\theta=y \sqrt{3}$
$\cot \theta=\sqrt{3}$
$\theta=30^{\circ}$
$ycot$ $\theta=y$
$\cot \theta=1$
$\theta=45^{\circ}$
Standard 11
Physics
