Important Formulas - Boats and Streams

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Important Formulas - Boats and Streams

1. Downstream

In water, the direction along the stream is called downstream.

2. Upstream

In water, the direction against the stream is called upstream.

3. Let the speed of a boat in still water be u km/hr and the speed of the stream be v km/hr, then

Speed downstream = (u + v) km/hr
Speed upstream = (u - v) km/hr.

4. Let the speed downstream be a km/hr and the speed upstream be b km/hr, then

Speed in still water $=\frac{1}{2}(a+b)$ km/hr
Rate of stream = $\frac{1}{2}(a-b)$ km/hr

Some more shortcut methods

5. Assume that a man can row at the speed of $x$ km/hr in still water and he rows the same distance up and down in a stream which flows at a rate of $y$ km/hr. Then his average speed throughout the journey

$=\frac{\text{Speed downstream × Speed upstream}}{\text{Speed in still water}}$ $=\frac{(x+y)(x-y)}{x}\text{km/hr}$

6. Let the speed of a man in still water be $x$ km/hr and the speed of a stream be $y$ km/hr. If he takes $t$ hours more in upstream than to go downstream for the same distance, the distance

$=\frac{(x^2-y^2)t}{2y}\text{km}$

7. A man rows a certain distance downstream in $t_1$ hours and returns the same distance upstream in $t_2$ hours. If the speed of the stream is $y$ km/hr, then the speed of the man in still water

$=y\left(\frac{t_2+t_1}{t_2-t_1}\right)\text{km/hr}$

8. A man can row a boat in still water at $x$ km/hr in a stream flowing at $y$ km/hr. If it takes him $t$ hours to row a place and come back, then the distance between the two places

$=\frac{t(x^2-y^2)}{2x}\text{km}$

9. A man takes $n$ times as long to row upstream as to row downstream the river. If the speed of the man is $x$ km/hr and the speed of the stream is $y$ km/hr, then

$x=y\left(\frac{n+1}{n-1}\right)$