Question

Two swimmers $A and B$ , one located on one side and other on the another side of a river, are situated at a distance $D$ from each other. The line joining them is making an angle $θ$ with the direction perpendicular to the flow. The speed of each swimmer with respect to still water is $u$ and speed of river flow is $v_{r}$ . Both $A and B$ start swimming at the same time in the direction parallel to line $A B$ towards each other and they keep on swimming in the same direction. Then,

(a) Find the time after which they meet.
(b) Find the speed of river $(v_{r})$ so that the path of the two swimmers with respect to the ground becomes perpendicular to each other.

Accepted Answer

(a) Given,
the magnitude of velocity of swimmer $A$ with respect to the water, $(v_{Aw}) = u$ ,
the magnitude of velocity of swimmer $B$ with respect to the water, $(v_{Bw}) = u$ ,
the velocity of water, $(v_{w}) = V_{r}$ .
The distance between point $A$ and $B = D$ .

The velocity of swimmer $A$ with respect to the swimmer $B, ({\vec{v}}_{AB}) = {\vec{v}}_{A} - {\vec{v}}_{B}$ ,
$\Rightarrow {\vec{v}}_{AB} = ({\vec{v}}_{Aw} + {\vec{v}}_{w}) - ({\vec{v}}_{Bw} + {\vec{v}}_{w})$
$\Rightarrow |v_{AB}| = |{\vec{v}}_{Aw} - {\vec{v}}_{Bw}| = u - (- u) = 2 u$ .

The time taken by $D$ to reach $θ$ ,

$(t) = \frac{distance between A B}{|{\vec{v}}_{BA}|}$
$\Rightarrow t = \frac{D}{2 u}$ .

(b) If with respect to the ground, both swimmers are moving in perpendicular direction, it means the dot product of their velocities with respect to the ground must be zero,

$\Rightarrow {\vec{v}}_{A} . {\vec{v}}_{B} = 0$
$\Rightarrow ({\vec{v}}_{Aw} + {\vec{v}}_{w}) . ({\vec{v}}_{Bw} + {\vec{v}}_{w}) = 0$
$\Rightarrow (- u \hat{AB} + v_{r} \overset{⏜}{i}) . (u \hat{A B} + v_{r} \overset{⏜}{i}) = 0$
$\Rightarrow - u^{2} (\hat{AB} . \hat{AB}) + v_{r}^{2} (\overset{⏜}{i} . \overset{⏜}{i}) = 0$
$\Rightarrow v_{r} = u$ .

Hence, the speed of the river must be equal to the speed of swimmers so that with respect to the ground, both swimmers appears to be moving perpendicularly.

Important Questions on Kinematics

Learn and Prepare for any exam you want !