River boat problem
RIVER – BOAT CONCEPT: In river boat concept VR = absolute velocity of river or velocity of river with respect to ground,
VB = absolute velocity of boat or velocity of boat with respect to ground and
VBR = velocity of boatman with respect to river or velocity of boat in still water.
Let us consider boatman starts from point A from one bank of river with velocity VBR as shown in figure. 
=
– 
Or, = + .
We can resolute all the velocities along x and y axis.
Absolute velocity of boat along x axis is VBx = – VBR sinθ+ VR = VR – VBRsinθ.
Absolute velocity of boat along y axis is VBy = VBR cosθ+ VR 
= VBR cosθ
If D is the width of the river then time to cross the river is t = 
= 
——- (i)
Drift or displacement of boatman along x axis when he reaches the other bank is x = VBxt = (VR – VBR sinθ)t
= 
—–(ii)
Conditions:(i) When the boat crosses the river in shortest interval of time: From equation (i), time should be minimum if
θ = 00 and that time is t = 
. I.e. boatman should drive the boat perpendicular to river (along AB).
(ii) When boatman wants to reach opposite point (point B) from where he started: From equation (ii), drift x = 0
Or, = 0.
So, 
= 
Or, sinθ = 
∴θ = 
()
Therefore, to reach point B, boatman should row the boat at an angle θ = () ) upstream from AB.
(iii) Shortest path travelled by boatman: Path travelled by boatman when he reaches the opposite shore is S = 
.
(a) When
<
: From equation (ii), drift x = 0 at θ = ().
So, minimum path travelled by boatman Smin = D.
(b) When >: From equation (ii), drift x is minimum when 
= 0
Or,
[] = 0
Or, [
– tanθ ] = 0
Or,
– 
θ = 0
Or,
= secθ
Or, sin θ=
∴θ = ().
Therefore minimum value of x is xmin = 
=
= 
. [Using equation (ii)]
Shortest path travelled by boatman is Smin = 
=
= 
.
EXAMPLE: The width of a river is D. A swimmer crosses the river and moves to the opposite point on the shore for time t1 when there is no current in the river. He takes time t2 to cross the river when river speed is v. Find the speed of river.
Let us consider the speed of swimmer in still water is u. so, D = ut1.
When speed of river is v, then actual speed of swimmer is 
So, D = t2.
Then, D2 = (
)
Or, D2 = (
– 
)
∴ v = D
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