The famous river-crossing problem

Classical Mechanics Level 3

A boat wants to cross a river of width $100\text{ m}$ from point A to C . The speed of stream current is $30\text{ ms}^{-1}$ and that of the boat in still water is $5\text{ ms}^{-1}$ . But due to the heavy force of water current the boat displaces to a point B.

If so find the value of the magnitude of displacement, ie value of AB.

Give your answer to 3 decimal places.

The answer is 608.27.

2 solutions

Ankur Shukla
Feb 19, 2017

Taking vertical motion perpendicular to stream and assuming time taken for complete trip is $t$ ,

$5\times t = 100$ that employs

$t = 20$

Again for horizontal motion,

$BC = 30\times 20$

$BC = 600$

So total displacement is Vector sum of AC and BC i.e.

$AB = \sqrt{100^2 + 600^2}$

that employs

$AB = 608.27625$

Aziz Alasha
Nov 26, 2016

It is well known from classical mechanics that :
Resultant velocity = sqrt. {900 + 25} =sqrt.925
Direction is determined by angle ♡ such that ;
Tan ♡ = 5/30 = 1/6 , sin ♡ = 1/sqrt.37 ,
The direction of the displacement vector will be the same as that of the resultant velocity.
There fore ;
Displacement = AB = 100 ÷ sin♡ = 100sqrt.37
AB = 608.300
QED.

Yes answer is incorrect. It will be 608.33... Butits 600.33...