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3-2.Motion in Plane
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A steam boat goes across a lake and comes back : $(i)$ on a quiet day when the water is still and $(ii)$ on a rough day when there is a uniform current so as to help the journey onward and to impede the journey back. If the speed of the launch, on both days, was same, the time required for the complete journey on the rough day as compared to that on the quiet day, will be
Aless
Bsame
Cmore
Dcannot be predicted
Solution
If $L$ be the length of the lake and the velocity of
boat is $V,$ time taken in going and coming back on a quiet day
$\mathrm{t}_{\mathrm{Q}}=\frac{\mathrm{L}}{\mathrm{V}}+\frac{\mathrm{L}}{\mathrm{V}}=\frac{2 \mathrm{L}}{\mathrm{V}}$ $…(i)$
Now, if $v$ is the velocity of air-current, then time
taken in going across the lake,
$t_{1}=\frac{L}{V+v}$ (as the current helps the motion and time taken in coming back,
$\mathrm{t}_{2}=\frac{\mathrm{L}}{\mathrm{V}-\mathrm{v}}($ as the current opposes the motion
So, $t_{R}=t_{1}+t_{2}=\frac{2 L V}{V^{2}-v^{2}}=\frac{2 L}{V\left[1-\left(\frac{v}{V}\right)^{2}\right]} \cdots(ii)$
Hence, from eqns. $(i)$ and $(ii),$
${\frac{\mathrm{t}_{\mathrm{R}}}{\mathrm{t}_{\mathrm{Q}}}=\frac{1}{1-(\mathrm{U} / \mathrm{V})^{2}}>1 \quad\left[\because 1-\left(\frac{\mathrm{v}}{\mathrm{V}}\right)^{2}<1\right]}$
${\mathrm{t}_{\mathrm{R}}>\mathrm{t}_{\mathrm{Q}}}$
boat is $V,$ time taken in going and coming back on a quiet day
$\mathrm{t}_{\mathrm{Q}}=\frac{\mathrm{L}}{\mathrm{V}}+\frac{\mathrm{L}}{\mathrm{V}}=\frac{2 \mathrm{L}}{\mathrm{V}}$ $…(i)$
Now, if $v$ is the velocity of air-current, then time
taken in going across the lake,
$t_{1}=\frac{L}{V+v}$ (as the current helps the motion and time taken in coming back,
$\mathrm{t}_{2}=\frac{\mathrm{L}}{\mathrm{V}-\mathrm{v}}($ as the current opposes the motion
So, $t_{R}=t_{1}+t_{2}=\frac{2 L V}{V^{2}-v^{2}}=\frac{2 L}{V\left[1-\left(\frac{v}{V}\right)^{2}\right]} \cdots(ii)$
Hence, from eqns. $(i)$ and $(ii),$
${\frac{\mathrm{t}_{\mathrm{R}}}{\mathrm{t}_{\mathrm{Q}}}=\frac{1}{1-(\mathrm{U} / \mathrm{V})^{2}}>1 \quad\left[\because 1-\left(\frac{\mathrm{v}}{\mathrm{V}}\right)^{2}<1\right]}$
${\mathrm{t}_{\mathrm{R}}>\mathrm{t}_{\mathrm{Q}}}$
Standard 11
Physics
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