# Boats and Streams Mathematics

Boats and Streams is just a logical extension of motion in a straight line. One or two questions are asked from this chapter in almost every exam. Here is discussion of some important facts and terminologies which will help you to make better understanding about this topic.

1. Still water: The water of a river or any other water body which is not flowing is known as still water.

2. Stream: It is the flowing water of a river which is moving at a certain speed.

3. Upstream: The boat or a swimmer moving against the stream is known as moving upstream i.e. against the flow of water.

4. Downstream: The boat or a swimmer moving along the stream is known as moving downstream i.e. along the flow of water.

Basic Concept
If direction of boat is same as direction of the stream, then it is known as DOWNSTREAM and if directions are opposite, then it is known as UPSTREAM. Following figure is representing the same:  i.e. if boat is moving with stream then it is known as Downstream and if opposite to stream, then it is Downstream.

Downstream Speed and Upstream Speed
In case of Downstream, as you can see the direction is same, speeds of stream and boat will be added to get Down stream speed.

If Speed of boat in still water = u km/hr

Speed of stream = v km/hr, then

Downstream Speed = (u+v) km/hr

Similarly, if I talk about upstream speed, as the direction of boat and stream is opposite, speed of both will be subtracted.

i.e. Upstream Speed = (u-v) km/hr

Study the following figure, notice the directions and try to remember this i.e.

If directions are same then speeds will be added and

If directions are opposite then speeds will be subtracted  Speeds of Boat and Stream if Downstream and Upstream Speeds are given

Speed of Boat = ½ (Downstream Speed + Upstream Speed)

Speed of Stream = ½ (Downstream Speed – Upstream Speed)

1. If the speed of the boat or swimmer is X km/hr and the speed of the stream is Y km/hr,

The speed of the boat or swimmer in the direction of the stream is known as speed downstream. It is given by;
Speed downstream= (X+Y) km/hr
And, the speed of the boat or swimmer against the stream is known as speed upstream. It is given by;
Speed upstream= (X-Y) km/hr

2. Speed of man or boat in still water is given by;

 =  3. Speed of the stream is given by;

 =  4. A man can row at a speed of X km/hr in still water. If the speed of the stream is Y km/hr and the man rows the same distance up and down the stream, his average speed for the entire journey is given by;    5. A man can row a boat in still water at X km/hr. If the stream is flowing at Y km/hr it takes him t hours more to row upstream than to row downstream to cover the same distance. The distance is given by;  6. A man can swim in still water at X km/hr. If the stream is flowing at Y km/hr it takes him t hours to reach a place and return back to the starting point. The distance between the place and the starting point is given by;  7. A boat or swimmer covers a certain distance downstream in t1 hours and returns the same distance upstream in t2 hours. If the speed of the stream is Y km/hr, the speed of boat or man in still water is given by;  8. A boat or swimmer takes K times as long to move upstream as to move downstream to cover a certain distance. If the speed of the stream is Y km/hr, the speed of the boat or man in still water is given by;  Problems with Solution

Example 1: Speed of boat in still water is 5 km/hr and speed of stream is 1 km/hr. find the downstream speed and upstream speed.

Solution: Given that, u = 5 km/hr

v = 1 km/hr

Downstream speed = u+ v km/hr

⇒ 5+ 1= 6 km/hr

Upstream speed = u -v km/hr

⇒ 5- 1 = 4 km/hr

Example 2: A man takes 3 hours to row a boat 15 km downstream of river and 2 hours 30 min to cover a distance of 5 km upstream. Find speed of river or stream.

Solution: We need to find speed of stream from downstream speed and upstream speed. See how we calculate it:

As you know, Speed = Distance/ Time

So, Downstream Speed = 15/3 = 5 km/hr

Upstream Speed = 5/2.5 = 2 km/hr

Now, as we have discussed, Speed of stream = 12 (Downstream Speed – Upstream Speed)

⇒Speed of stream = ½ (5-2)

⇒3/2 = 1.5 km/hr

Example 3: A man can row 7 km/hr in still water. If in a river running at 2 km/hr, it takes him 50 minutes to row to his place and back, how far off is the place?

Solution: Given, u = 7 km/hr

v = 2 km/hr

From u and v, we can calculate downstream speed and upstream speed.

Downstream Speed = (u + v) = 7+2 = 9 km/hr

Upstream Speed = (u-v) = 7-2 = 5 km/hr

Now, we need to find DISTANCE and time is given,

Time = Distance / Speed

Let required distance = x km

Time taken in downstream + Time taken in upstream = Total Time

⇒ ?/9 + ?/5 = 50/60 (50 minutes = 50/60 hrs)

⇒Calculating the above equation: x = 2.68 km