Scalar & vector problem

Q - :
Two men A and B can swim in still water, at the rate of 5km/hr. The man A swims upstream a distance half km equal to width of river and B swims straight across the river and back again. Assuming that the velocity of river is 3 km/hr, find the difference in the time taken by them.

Sol - :


Here, velocity of man is, v = 5 km/hr

Velocity of river is, u = 3 km/hr, along QP.

Swimmer - A


Velocity of swimmer A w.r.t. bank while going from P to Q = v - u

Velocity of swimmer A w.r.t. bank while coming back from Q to P = v + u

Time taken by swimmer A to make the round trip is,

t' = d/v-u + d/v+u
   = 2vd / v^2 - u^2
   = 2*5*0.5 / 5^2 - 3^2
   = 5/16 hr
   = 18 min 45 sec

Swimmer - B

To cross the river straight across man will dive at angle @ given by,

Sin @ = u / v = 3/5

Therefore, time taken by man B to make the round trip is - :

t'' = 2d / vcos @
    = 2*0.5 / 5*0.8
    = 1/4 hr
    = 15 min

Difference of time taken by two swimmers = t' - t''
 = 18 min 45 sec - 15 min
 = 3 min 45 second