10. A river is flowing from west to east at a speed of 5m per minute. A man on the south bank of the river, capable of swimming at 10m per minute in still water wants to swim across the river to a point directly opposite in the shortest time. He should then swim in a direction-(a) 60° west of north(b) 30°east of north(c) 30°west of north(d) 60°east of north

The answer for this question is (c) 30 west of north.

Solution:

Contradiction for (b),(d)

Comparing situation (a) with situation (d)

In situation (a) and (d) the vertical component of man velocity in north direction are equal thus in both situation he ll cross the  river in equal time.

But in situation (d) the horizontal component of velocity of man is in direction of river velocity so he will go away frm  the point on the other side. On the other hand in situation (a) the horizontal componet of velocity of man opposes the velocity the river and as such he will be closer to the point on other side after crossing the river. So (d) is clearly not take the shortest time

Similar situation arises when we compare option (b) and (c). (b) will not take the shortest time.

Contradiction for (a)

Comparing situation (a) with situation (c)

In situation (c), the component of man velocity in north direction is more than situation (a) so in situation (c) he will take lesser time to cross the river.

Moreover in situation (c), the horizontal component of velocity is such that it just  balances the velocity of the river and man will land on point on the other side after crossing the river.

In situation  (d) the horizontal velocity component is more than river velocity so he will land to the west of the point  and had to transverse the remainig distance on foot