Formulas
1.If speed of boat or swimmer is x km/h and the speed of stream is y km/h then,
Speed of boat or swimmer upstream = (x - y) km/h
Speed of boat or swimmer downstream = (x + y) km/h
2.Speed of boat or swimmer in still water =$\dfrac{1}{2}\left(Downstream\:Speed+Upstream\:Speed\right)$
3.Speed of stream =$\dfrac{1}{2}\left(Downstream\:Speed-Upstream\:Speed\right)$
4.A man can row certain distance downstream in t1 hours and returns the same distance upstream in t2 hours. If the speed of stream is y km/h, then the speed of man in still water is given by
$y\left(\dfrac{t_2+t_1}{t_2-t_1}\right)km/h$
5.A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes him t hours to row to a place and come back, then the distance between two places is given by
$\dfrac{t\left(x^2-y^2\right)}{2x}km$
6.A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes t hours more in upstream than to go downstream for the same distance, then the distance is given by
$\dfrac{t\left(x^2-y^2\right)}{2y}km$
7.A man can row in still water at x km/h. In a stream flowing at y km/h, if he rows the same distance up and down the stream, then his average speed is given by
$\dfrac{\left(x^2-y^2\right)}{x}km/h$
Boats and Streams
Still Water:
If the water is not moving then it is called still water.
Stream:
Moving water of the river is called stream.
Upstream:
If a boat or a swimmer moves in the opposite direction of the stream then it is called upstream.
Downstream:
If a boat or a swimmer moves in the same direction of the stream then it is called downstream.
1.If speed of boat or swimmer is x km/h and the speed of stream is y km/h then,
Speed of boat or swimmer upstream = (x - y) km/h
Speed of boat or swimmer downstream = (x + y) km/h
Example:
If the speed of boat in still water is 10 km/hr & the speed of stream is 3 km/hr,then find Downstream speed and Upstream speed?
Speed of boat in still water x = 10 km/hr
Speed of stream = y = 3 km/hr
Then,
Downstream speed=(10+3)=13km/hr
Upstream speed=(10-3)=7km/hr
Exercise:
Speed of boat in still water = x = 20 km/hr
Speed of stream = y = 5 km/hr
Then,
Downstream speed=(20+5)=25km/hr
Upstream speed=(20-5)=15km/hr
2.Speed of boat or swimmer in still water is given by
$\dfrac{1}{2}\left(Downstream\:Speed+Upstream\:Speed\right)$
Example:
A boat takes 6 hours to cover 36 km downstream and 8 hours to cover 32 km upstream. Then the speed of the boat in still water is?
Speed downstream = $\dfrac{Distance}{Time}$=$\dfrac{36}{6}$=6km/hr
Speed upstream = $\dfrac{Distance}{Time}$=$\dfrac{32}{8}$=4km/hr
Speed of stream =$\dfrac{1}{2}\left(Downstream\:Speed+Upstream\:Speed\right)$
Required speed =$\dfrac{1}{2}\left(6+4\right)km/hr $
=$\dfrac{1}{2}\times 10$
=5km/hr
Exercise:
Speed of stream =$\dfrac{1}{2}\left(Downstream\:Speed+Upstream\:Speed\right)$
Speed downstream = 20 km/hr
Speed upstream = 15 km/hr
Required speed =$\dfrac{1}{2}\left(20+15\right)km/hr$
=$\dfrac{1}{2}\times 35$
=17.5km/hr
3.Speed of stream is given by
$\dfrac{1}{2}\left(Downstream\:Speed-Upstream\:Speed\right)$
Example:
A man swims 12 km downstream and 10 km upstream. If he takes 2 hours each time, what is the speed of the stream?
Speed downstream = $\dfrac{Distance}{Time}$=$\dfrac{12}{2}$=6km/hr
Speed upstream = $\dfrac{Distance}{Time}$=$\dfrac{10}{2}$=5km/hr
Speed of stream =$\dfrac{1}{2}\left(Downstream\:Speed-Upstream\:Speed\right)$
Required speed =$\dfrac{1}{2}\left(6-5\right)km/hr$
=$\dfrac{1}{2}\times 1$
=0.5km/hr
Exercise:
Speed downstream = $\dfrac{Distance}{Time}$=$\dfrac{36}{6}$=6km/hr
Speed upstream = $\dfrac{Distance}{Time}$=$\dfrac{32}{8}$=4km/hr
Speed of stream =$\dfrac{1}{2}\left(Downstream\:Speed-Upstream\:Speed\right)$
Required speed =$\dfrac{1}{2}\left(6-4\right)km/hr$
=$\dfrac{1}{2}\times 2$
=1km/hr
4.A man can row certain distance downstream in t1 hours and returns the same distance upstream in t2 hours. If the speed of stream is y km/h, then the speed of man in still water is given by
$y\left(\dfrac{t_2+t_1}{t_2-t_1}\right)km/h$
Example:
A man goes certain distance against the current of the stream in 2 hour and returns with the stream in 20 minutes. If the speed of stream is 4 km/h then how long will it take for the man to go 4 km in still water?
Let’s say t1 = 20 minutes = 0.33 hours and t2 = 1 hours
Y = 4, then man’s speed in still water
speed of man in still water=$y\left(\dfrac{t_2+t_1}{t_2-t_1}\right)$
=$4\left(\dfrac{1+0.33}{1-0.33}\right)$
=$\dfrac{4\times1.33}{0.67}$
=7.94km/h
So man’s speed is 7.94 km/h in still water.
Now, time taken by the man to row 4 km in still water
=$\dfrac{4\times1}{7.94}$
=0.504hours
=30.23minutes
Exercise:
Speed of boat in still water=$y\left(\dfrac{t_2+t_1}{t_2-t_1}\right)km/h$
With the given parameters , y = 6 km/hr, t1 = 3 hrs, t2 = 2 hrs
We can find, Speed of boat in still water (x) = $6\left(\dfrac{3+2}{3-2}\right)$
= 30 km/hr
5.A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes him t hours to row to a place and come back, then the distance between two places is given by
$\dfrac{t\left(x^2-y^2\right)}{2x}km$
Example:
A man can row 4 km/h in still water. When the water is running at 2 km/h, it takes him 2 hours to go to a place and come back. What is the distance between that place and man’s initial position?
Let’s say x = 4 km/h = man’s speed in still water.
y = 2 km/h = water’s speed.
t = 2, so
Distance=$\dfrac{t\left(x^2-y^2\right)}{2x}km$
=$\dfrac{2\left(4^2-2^2\right)}{2\times 4}$
=3km
Exercise:
Distance between two places =$\dfrac{t\left(x^2-y^2\right)}{2x}$
Given parameters are:
Speed of boat (x) = 6 km/hr
Speed of water (y) = 4 km/hr
Time taken by the boat to go & return back = t = 3 hrs
To find the distance between the place & initial position of boat (i.e. between two places), we have
Distance between two places =$\dfrac{t\left(x^2-y^2\right)}{2x}$
=$\dfrac{3\left(6^2-4^2\right)}{2\times6}$
=$\dfrac{3\left(36-16\right)}{12}$
=$\dfrac{\left(3\times20\right)}{12}$
=5km/hr
6.A man can row in still water at x km/h. In a stream flowing at y km/h, if it takes t hours more in upstream than to go downstream for the same distance, then the distance is given by
$\dfrac{t\left(x^2-y^2\right)}{2y}km$
Example:
A man can row 4 km/h in still water. The water is running at 2 km/h. He travels to a certain distance and comes back. It takes him 2 hours more while travelling against the stream than travelling with the stream. What is the distance?
Let’s say x = 4 km/h = man’s speed in still water.
y = 2 km/h = water’s speed.
t = 2, so
Distance=$\dfrac{t\left(x^2-y^2\right)}{2y}km$
=$\dfrac{2\left(4^2-2^2\right)}{2\times2}$
=6km
Exercise:
Distance =$\dfrac{t\left(x^2-y^2\right)}{2y}km$
Given parameters are:
Speed of a boat in still water x= 10 km/hr
Speed of running water y= 4 km/hr
Required time t= 4 hrs to travel upstream more than downstream
Therefore, we obtain,
Distance =$\dfrac{t\left(x^2-y^2\right)}{2y}$
=$\dfrac{4\left(10^2-4^2\right)}{2\times4}$
= 42 km
7.A man can row in still water at x km/h. In a stream flowing at y km/h, if he rows the same distance up and down the stream, then his average speed is given by
$\dfrac{\left(x^2-y^2\right)}{x}km/h$
Example:
Speed of boat in still water is 9 km/h and speed of stream is 2 km/h. The boat rows to a place which is 47 km away and comes back in the same path. Find the average speed of boat during whole journey.
Let’s say x = 9 km/h = speed in still water
Average Speed=$\dfrac{\left(x^2-y^2\right)}{x}km/h$
=$\dfrac{\left(9^2-2^2\right)}{9}km/h$
=8.55km/h
Y = 2 km/h = speed of stream
Exercise:
Given Parameters :
Speed of boat in still water = x = 10 km/hr
Speed of stream = y = 3 km/hr
We have,
Average Speed=$\dfrac{\left(x^2-y^2\right)}{x}km/h$
=$\dfrac{\left(10^2-3^2\right)}{10}km/h$
=$\dfrac{\left(100-9\right)}{10}km/h$
=9.1km/hr