# Boats and Stream

u – Speed of boat/man

v – Speed of water

Speed of boat along the river (Downstream) a = u + v

Speed of boat against the river (Upstream) b = u – v

Mostly speed if boat greater than speed of water

Speed of boat in still water u = ½(a + b)

Speed of water current v = ½(a – b)

Distance = time (u^{2 }– v^{2})/ 2u

1) A man can row 18km/hr in still water. speed of the man in downstream is thrice the speed in upstream. Find the rate of stream.

Solution:

Let, Speed of man in upstream a = a

Speed of man in downstream b = 3a

Speed of man in still water u = ½(a + b)

Speed of man in still water = ½(3a + a) = 2a

We know speed of man in still water = 18

So, a = 9

Rate of stream = ½(27 – 9) =9 km/hr

2) A boat can cover certain distance in downstream in 1hr. and it takes 1½hr to cover same distance in upstream. If speed of the stream is 3kmph, then what will be the speed of boat?

Solution:

let speed of boat in still water be x

speed in downstream = x + 3

speed in upstream = x – 3

Since boat covered same distance in upstream and downstream,

(x + 3)*1 = (x – 3)*(3/2)

speed in downstream x= 15 kmph

3) A man can row three quarter of a km against the stream in 11¼ min and down the stream in 7½min. what is the speed of man in still water.

Solution:

Speed of man in Upstream = ((¾)/(45/4)) *60

= 4 kmph

Speed of man in Downstream = ((¾)/(15/4))* 60

= 12 kmph

Speed of man in still water u = ½(a + b)

Speed of man in still water = (12 + 4)*½ = 8 kmph

4) A streamer takes 3hr to cover a distance of 24km upstream, if the rate of stream is 3 kmph. Then find the speed of streamer in still water.

Solution:

Upstream speed = 24/3 =8kmph

Rate of stream = 3kmph

Speed of streamer – rate of stream = upstream

Speed of streamer = 11kmph

5) The distance between two points is 36km. A boat rows in still water at 6kmph, it takes 8hr less to cover dist in downstream in comparison to that in upstream. Find the rate of stream.

Solution:

Time = distance * speed

Difference between time taken to cover upstream and downstream is 8hr

(36/(6-x)) – (36/(6+x)) = 8

36(6+x) – 36(6-x) = 8(36-x^{2})

9x = 36- x^{2}

x^{2}+9x-36 = 0 (to find value of x use quadratic equation technique)

(x+12)(x-3)=0

Speed cannot be negative so x = 3kmph

Rate of stream = 3kmph