Two boats, each with velocity `8km*h^(-1)` , attempt to cross a river of width 800 m. The velocity of river current is `5 km*h^(-1)` . One of the boats crosses the river following the shortest path and the other follows the route in which the time taken is minimum. If they start simultaneously, what would be the time difference between their arrivals at the other bank ?

Explanation

In order to follow the shortest path , the boat should set itself at a particular angle with the current such that the boat,s resultant velocity is perpendicular to the direction of the current [Fig.2.52 (a)]. Hence the magnitude of the resultant velocity of the boat,

`w=sqrt(v^2-u^2)`
and the time required to cross the river along the shortest path,
`t_1=x/w=x/(sqrt(v^2-u^2))=(0.8)/(sqrt(8^2-5^2))=0.128h =7.69 ` min
[x=800 m =0.8km, v=`8 km *h^(-1) and 5 km *h^(-1)`]
To cross in minimum time , the boat should travel at right angles to the current [Fig.2.52(b)] and the time required is
`t_2=x/v=(0.8)/(8) =0.1 h =6 `min
`therefore` Required time difference,
`t_1-t_2=7.69-6=1.69 ` min