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 Downstream 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 = 1/2 (Downstream Speed + Upstream Speed)
Speed of Stream = 1/2 (Downstream Speed — Upstream Speed)
Problems with Solution
Example1: 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
Example2: A man takes 3 hours to row a boat 15km 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 I 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 i have discussed, Speed of stream = 1/2 (Downstream Speed — Upstream Speed)
⇒Speed of stream = 1/2 (5-2) ⇒3/2 = 1.5 km/hr
Example3: A man can row 7km/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? *Important Question*
Solution: Given, u = 7km/hr v = 2km/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 ⇒ (x/9) + (x/5) = (50)(60)........................ 50 minutes = (50)(60) hrs
⇒Calculating the above equation: x = 2.68km