Question

A boat covers certain distance between 2 spots in a river taking t1 hrs going downstream and t2 hrs going upstream.what time will be taken by boat to cover same distance in still water?

Answer 1

`t_"still" = (2t_1t_2)/(t_1+t_2)`

Explanation:

We're asked to find the expression for the time it takes the boat to travel across the river in still water.

I'll try and show this with a little example:

Suppose a boat is traveling at a constant speed `v` (without a current), and the current's speed is `u`

Going upstream, the true speed of the boat would be

`v_"up" = v+u`

And going downstream, we have

`v_"down" = v-u`

We have our general speed equation:

`v = s/t`

Or in these two cases

`v+u = s/(t_1)`

`v-u = s/(t_2)`

(the distance across the river `s` is constant)

The quantity `u` (the current's speed) is constant for both equations, so let's solve both equations for `u` and set them equal to each other:

`u = s/(t_1) - v`

`u = v - s/(t_2)`

So

`color(red)(s/(t_1) - v = v - s/(t_2)`

`2v = s/(t_1) + s/(t_2)`

Here, the speed `v` is the speed of the boat without the current, and is thus equal to

`v = s/(t_"still")`:

`(2s)/(t_"still") = s/(t_1) + s/(t_2)`

`2/t_"still" = 1/(t_1) + 1/(t_2)`

`t_"still" = 2/(1/(t_1) + 1/(t_2))`

Get the denominator's fractions into a common denominator:

`t_"still" = 2/((t_2 + t_1)/(t_1t_2))`

or

`color(blue)(ulbar(|stackrel(" ")(" "t_"still" = (2t_1t_2)/(t_1 + t_2)" ")|)`