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Motion in a Plane — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsMotion in a Plane3 Marks
Question
A man can swim at the rate of $5km/ h$ in still water. A river $1km$ wide flows at the rate of $3km/ h$. A swimmer wishes to cross the river straight.
  1. Along what direction must he strike?
  2. What should be his resultant velocity?
  3. How much time he would take to cross?
✓

Answer



Width of the river, $d = 1km$
Velocity of swimmer, $v_s = 5km/ h$
Velocity of water flowing in river,
$v_r = 3km/ h$ along OQ.
  1. The swimmer wants to cross the river straight if the resultant velocity of the river flow and swimmer acts perpendicular to the direction of the river flow i.e. along OP. This will be so if the swimmer moves making an angle a with upstream i.e. along OR.
​​​​​​​But, $\alpha+\theta=90^\circ$ or $\theta=90^\circ-\alpha$
From $\triangle\text{OPR},$ we have
$\sin\theta=\sin(90^\circ-\alpha)=\cos\alpha$
$=\frac{\text{RP}}{\text{RO}}=\frac{3}{5}=0.6$
$\therefore\ \cos\alpha=\cos53^\circ8'$
$\Rightarrow\ \alpha=53^\circ8'\ \text{upstream}$
  1. The resultant velocity along OP is given by
​​​​​​​$\text{v}=\sqrt{\text{v}^2_\text{s}-\text{v}^2_\text{r}}=\sqrt{5^2-3^2}=4\text{km/h}$
  1. Time taken by swimmer to cross the river
​​​​​​​$\text{t}=\frac{\text{d}}{\text{v}}=\frac{1}{4}=0.25\text{h}=15\text{ min}$

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