Consider the problem:
A river flows west to east at a speed of 5m per minute. A man on the south bank can swim at the rate of 10m per minute in still water. In which direction should he swim so as to cross the river
1) in the smallest possible time
2) along the shortest path
Solution
Take the normal NSEW directions (N up, E right, etc). Assume XY axes such that+ve X is along east and +Y along north. Let the origin of the coordinate system be at a point on the south shore from where the man starts.
Speed of man wrt water, vMW = 10 m/min
Velocity of water wrt to earth, vWE = 5 i m/min
Let velocity of man wrt eath be vME.
Let the width of the river be d.
(a) Let the man try to swim in a direction making 
with the X axis.
vMW = 10(cos i + sin j)
vME = vMW + vWE = (10cos + 5) i + 10sin j
Time taken by the man to cross, T = d/10sin
T is minimum if sin is maximum, ie, = 
/2
Hence the man should swim perpendicular to the current if he is to reach in shortest possible time.
(b) The shortets possible path is pependicular to the river. For the man to follow this path, vME must not have any velocity component along X axis.
10cos + 5 = 0
= 1200
Hence the man should swim in a direction making 1200 the +ve X direction
Moderation Team
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