Question

A steamer taken m minutes to go a mile downstream and n minutes to go a mile upstream. Find the speed of the
current and of the steamer relative to it

Answer 1

Let streamer speed be $v_{b}$ , and current speed be $v_{c}$

For 1st case :

Since does steamer is going downstream its speed is enhanced by the speed of the current in the water. As a result , the net time taken to cover a distance of 1 mile will be much less as compared to do when it goes upstream.

$\dfrac{1}{v_{c} + v_{b}} = m$

$= > v_{c} + v_{b} = \dfrac{1}{m} \: .....(1)$

For 2nd case :

Since the streamer is going upstream its speed is being imposed by the speed of current in the water and hence the time taken will be much more.

$\dfrac{1}{v_{b} - v_{c}} = n$

$= > v_{b} - v_{c} = \dfrac{1}{n} \: ........(2)$

Solving the 2 equations, we get :

$1) \: \: \: v_{b} = \dfrac{m + n}{2mn} \\ \\ 2) \: \: \: v_{c} = \dfrac{n - m}{2mn}$

Relative velocity will be $\Delta \: v$

By relative velocity we mean the difference of downstream an upstream velocity.

$\boxed{\Delta v = v_{b} - v_{c} = \dfrac{1}{n} }$