Boats and Streams

Distance = Speed × Time 1km/hr = 5/18 m/s 1 m/s = 18/5 km/hr Speed α Ditance If the speed is doubled, distance covered in

a given time is also doubled. Speed α 1/ Time If the speed is doubled, time taken to cover

a distance would half.

General Concept

BOATS: Downstream motion of a boat is its motion

in the same direction as the flow of the river.

Upstream motion of a boat is its motion in the opposite direction as the flow of the river.

Speed of the river is the speed with which rivers flows

Speed of the boat in the still water, is the speed at which the boat would be moving, if the water is still.

Boat is moving along with flow of river (B), so water stream (S) helps the boat to move faster.

It is same like “A and B work together”. So their speed will increase and we can do addition.

Hence Downstream speed = speed of Boat PLUS speed of stream (B+S)

Down Stream

Boat is moving against the direction of river. It is same like “Pipe A can fill the tank in 1 hours while Pipe B can empty the tank in 2 hours”

In short they work against each other, hence final speed is decreases so we’ve to subtract. (B-S)

Upstream speed = Speed of Boat MINUS speed of stream (B-S)

Up Stream

Points to Remember Instead of boats in water, it could be a swimmer or a cyclist cycling

against or along the wind.

When the speed of the boat is given, it is the speed of the boat in still water.

If b = Speed of boat in still water s = Speed of stream or water u = Speed of boat upstream d = Speed of boat downstreamThen,1. u = b-s2. d = b+s3. b = ½(u+d)4. s = ½(d-u)

Eg 1: A boat travels 36 km upstream in 9 hours and 42 km downstream in 7 hours. Find the speed of the boat in still water and the speed of the water current ?

CASE # 1 : Basics

Ans: Upstream speed of the boat = 36/9 = 4 kmph

Downstream speed of the boat = 42/ 7 = 6kmph.

Speed of the boat in still water = (6+4) / 2.

= 5 kmph

Speed of the water current = (6-4) /2

= 1 kmph

Solution

Q. A man can rows 27 km down stream and 18 km upstream taking 3 hr each time. What is the velocity of the current?

Q. If a man rows at 6 km/hr in still water and 4.5 k/hr against the current, then his rate of rowing along the current is

Eg : A man can row at 10 kmph in still water. If it takes a total of 5 hours for him to go to a place 24 km away and return, then find the speed of the water current ?

Ans: Let the speed of the water current be y kmph.

Upstream speed = (10- y) kmph

Downstream speed = (10+y) kmph

Total time = (24/ 10+y) + ( 24/10-y) = 5

Hence, 480/ (100-y2 ) = 5

480= 500-5y2

5y2= 20

y2= 4

y = 2 kmph.

Solution

A man can row x kmph in still waters. If in a stream which is flowing at y kmph, it takes him z hrs to row from A to B and back (to a place and back), then

The distance between A and B = z ( x² - y²) / 2x.

In the previous case, distance between A and B, time taken by the boat to go upstream and back again to the starting point, speed of the stream are given; then the speed of the boat in still waters can be obtained using the above given formula.

Case # 2: Total Time for upstream and downstream is given collectively.

A man can row 6 kmph in still water. When the river is running at 1.2 kmph, it takes him 1 hour to row to a place and back. How far is the place?

Ans: Required distance =[1 x ( 6² -( 1.2) ²)] /12 kmph

= (36 - 1.44) / 12

= 2.88 km.

Solution

Q. A stream runs at 1 km/hr. A boat goes 35 km upstream and backs again in 12 hr. The speed of the boat in still water is….

A man rows a certain distance downstream in x hours and returns the same distance in y hrs. If the stream flows at the rate of s kmph then,

The speed of the man in still water = s(x+y) / ( y-x) kmph.

A man rows a certain distance downstream in x hours and returns the same distance in y hours. If the speed of the man in still water m kmph, then

Speed of the stream = m (y-x) / (x+y) kmph.

Case # 3: Time taken for upstream and Down stream is given seprately

Q. Ramesh can row a certain distance downstream in 6 hours and return the same distance in 9 hours. If the stream flows at the rate of 3 kmph. Find the speed of Ramesh in still water?

Q. Ramesh can row a certain distance downstream in 9 hours and returns the same distance in 6 hours. If the speed of Ramesh in still water is 12 kmph. Find the speed of the stream?

Q. A river runs at 2 km/hr. If a man takes twice as long to row up the river as to row down, the speed of the man in still water is…