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3-2.Motion in Plane
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A missile is fired for maximum range at your town from a place $100\, km$ away from you. If the missile is first detected at its half way point, how much warning time will you have ? (Take $g = 10\, m/s^2$)
A$100\,s$
B$100\sqrt 2 \,s$
C$200\,s$
D$\frac{{100}}{{\sqrt 2 }}\,s$
Solution
For maximum range $\theta=45^{\circ}$ $\mathrm{R}=100 \mathrm{km}=10^{5} \mathrm{m}(\text { maximum })$
$\mathrm{R}=\frac{\mathrm{u}^{2}}{\mathrm{g}} \quad\left(\because \theta=45^{\circ}, \quad \therefore \sin 2 \theta=1\right)$
$\Rightarrow \mathrm{u}^{2}=\mathrm{g} \mathrm{R} \Rightarrow \mathrm{u}=\sqrt{\mathrm{gR}}$
Time of flight $\mathrm{T}=\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}}$
After top point, time remaining $\mathrm{t}=\frac{\mathrm{T}}{2}$
$\Rightarrow \mathrm{t}=\frac{\mathrm{u} \sin 45^{\circ}}{\mathrm{g}}$
$=\frac{\sqrt{\mathrm{gR}}}{\mathrm{g}} \times \frac{1}{\sqrt{2}}$
$=\sqrt{\frac{\mathrm{R}}{\mathrm{g}}} \cdot \frac{1}{\sqrt{2}}=\sqrt{\frac{10^{5}}{10}} \times \frac{1}{\sqrt{2}}$
$=\frac{100}{\sqrt{2}} \mathrm{s}$
$\mathrm{R}=\frac{\mathrm{u}^{2}}{\mathrm{g}} \quad\left(\because \theta=45^{\circ}, \quad \therefore \sin 2 \theta=1\right)$
$\Rightarrow \mathrm{u}^{2}=\mathrm{g} \mathrm{R} \Rightarrow \mathrm{u}=\sqrt{\mathrm{gR}}$
Time of flight $\mathrm{T}=\frac{2 \mathrm{u} \sin \theta}{\mathrm{g}}$
After top point, time remaining $\mathrm{t}=\frac{\mathrm{T}}{2}$
$\Rightarrow \mathrm{t}=\frac{\mathrm{u} \sin 45^{\circ}}{\mathrm{g}}$
$=\frac{\sqrt{\mathrm{gR}}}{\mathrm{g}} \times \frac{1}{\sqrt{2}}$
$=\sqrt{\frac{\mathrm{R}}{\mathrm{g}}} \cdot \frac{1}{\sqrt{2}}=\sqrt{\frac{10^{5}}{10}} \times \frac{1}{\sqrt{2}}$
$=\frac{100}{\sqrt{2}} \mathrm{s}$
Standard 11
Physics
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