River and Boat problem

How to solve problems of river and boat 

Main case 

1. To cross the river with the shortest possible path. 

2. To cross the river with the minimum possible time.

General info:

 Vb = velocity of boat in still water or (actual maximum possible velocity of boat)

 Vr = velocity of river w.r.t ground

   basic formula "SPEED = DISTANCE /TIME"

CASE 1 "To cross the river with the shortest possible path." 

why at an angle ?????

If you wish to cross a river and arrive at a dock that is directly across from you, but the river's current will tend to carry you downstream . Means to remove the effect of rivers flow ,speed or compensate that , you must steer the boat at an angle. 

Example :

say 
Vr = 4 m/s
V = Vb =5m/s
width of river = 100 m
then using above equations 
 Vcos(a) = Vr =>  5*cos(a) = 4 m/s
 => cos(a) = 4/5 

 => (a) = 37 degree 

so, Vsin(a) = 5*sin(37) = 5 * (3/5) m/s= 3m/s

T = time taken to cross river =  distance/time = w/Vsin(37) = 100/3= 33.33 seconds  
 required answer = 33.33 secconds.........................................(A)

2. To cross the river with the minimum possible time.

           

                                        time taken , Tmin =  W/ Vb

              Drift (due to current velocity) = Tmin*Vr

• To move with shortest time boat has to keep maximum of its Velocity in vertically direction,  i .e Vb .
• The river will still keep pushing boat toward its current direction and will shift its position from A to B.
• Here this shifting is named as drift .

Example :

say 

Vr = 4 m/s

V = Vb =5m/s

width of river = 100 m

then using above equations to cross the river with minimum possible time  

 T min = W/Vb = 100/5 = 20 seconds

 drift = Vr*Tmin = 4*20= 100 m 

T = time taken to cross river = 20 seconds  

 required answer = 20 secconds.........................................(B)

if we compare  equation (A) & (B) we can see that in (B) time was minimum.