Home
Class 12
PHYSICS
A man wishes to cross a river of width 1...

A man wishes to cross a river of width 120 m by a motorboat. His rowing speed in still water is `3 m//s` and his maximum walking speed is `1m//s`. The river flows with velocity of `4 m//s`.
(a) Find the path which he should take to get to the point directly opposite to his starting point in the shortest time.
(b) Also, find the time which he takes to reach his destination.

Text Solution

Verified by Experts

First you should check whether the simmer can reach the point b directly opposite to his staring point A along the line AB or not.
The velocity v of swinmmer relative to water is 3m/s which is less than the velocity of river flow ( u= 4 m/s) . In this condition , it is not possible to have any angle `theta` with the AB for which the resultant of ` vecv and vecu` is along AB. As u gt v, the swimmer will reach a point C somewhere to the right of point B for any angle `theta` he choose to start. After reaching that point C, he has to walk along the bank from C to B with walking speed ` v_(0)` ( = 1 m/s) . Let him take a time ` t_(1)` to reach the point C and then a time ` t_(2)` to walk to B from C. Now the angle `theta` he has to choose in such a way that the total time takent `(t_(1) + t_(2))_(0)` to reach B is minmum.
(a) ` t_(1) = d/v_(y) = d/( v cos theta) and tan alpha = ( u - v sin theta)/(v cos theta)`
` t_(2) = BC/v_(0) = (d tan alpha)/v_(0)`
Total time taken ` t= t_(1) + t_(2)`
` Rightarrow t = d/( v cos theta) + d/v_(0) ( ( u - v sin theta)/(v cos theta))`
For minimum value ` t , (dt)/(dtheta)=0 `


` Rightarrow (dt)(dtheta) = d/v sec theta tan theta + ( ud)/( v_(0)v) sec theta tan theta = d/(v_(0)) sec^(2)0=0`
` Rightarrow tan theta ( 1 + u /v_(0)) = v/v_(0) sec theta , or sin theta = v/( u + v_(0))`
` Rightarrow theta = sin^(-1) ( u/(u +v_(0))) = sin^(-1) (3/(4 +1)) = sin^(-1) (3/5)`
` Rightarrow ` the swimmer should start a angle ` , theta = sin(-1) (3/5) ` with AB.
(b) The total time taken is
` t = d/( v cos theta) + d/v_(0) ( (u-vsintheta)/(v cos theta)) = 120/(3 (4/5)) + 120/1 ((4 - 3xx (3/5))/(3xx (4/5))) = 160 s`
Promotional Banner

Topper's Solved these Questions

  • MOTION IN A PLANE

    AAKASH INSTITUTE|Exercise llustration 1 :|1 Videos

  • MOTION IN A PLANE

    AAKASH INSTITUTE|Exercise Try yourself|48 Videos

  • MOCK_TEST_17

    AAKASH INSTITUTE|Exercise Example|15 Videos

  • MOTION IN A STRAIGHT LINE

    AAKASH INSTITUTE|Exercise ASSIGNMENT (SECTION - D)|15 Videos

Similar Questions

Explore conceptually related problems

A man wants to cross a river 500 m wide. His rowing speed in still water is 3 kmph. The river flows at a speed of 2 kmph. If the man's walking speed on the shore is 5kmph, (a) find the path he should take to get to the point directly opposite to his starting point in the shortest time. (b) How long does it take to get there?

A swimmer can swim with speed 12 m/s in still water. Speed of river is 6 m/s. Find angle at which he should swim with downstream so that he reaches directly opposite point on other side

The rowing speed of a man relative to water is 5 kmh^(-1) and the speed of water flow is 3 kmh^(-1) . At angle to the river flow should he head if he wants to reach a point on the other bank, directly opposite to starting point ?

The velocity of a boat in still water is 10 m//s .If water flows in the river with a velocity of 6 m//s what is the difference in times taken to cross the river in the shortest path and the shortest time.The width of the river is 80 m .

A man is crossing a river flowing with velocity of 5m//s . He reaches a point directly across at a distance of 60m in 5sec. his velocity in still water should be

A man is crossing a river flowing with velocity of 5m//s . He reaches a point directly across at a distance of 60m in 5 sec. His velocity in still water should be

Velocity of boat in still water is 13 m/s. If water flows ina river with a velocity of 5 m/s. What is the difference in times taken by him to cross the river in the shortest path and in the shortest time. If the width of the river is 10 m.

A man can swim in still water with speed v . He wants to cross a river of width d that flows with speed u and reaches the point directely opposite to his starting point. (a) In which direction should he try to swim (i.e., find the angle the makes with the river flow)? (b) much time will he take to cross the river ?