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A river is flowing from west to east at speed of 5 m/minute . A man on south bank of river, capable of swimming at 10 m/min in still water, wants to swim across river in shortest time. He should swim in direction a)due north. b)30 east of north. c) 30 north of west. d) 60 east of north. Justify you answer.

A river is flowing from west to east at speed of 5 m/minute . A man on south bank of river, capable of swimming at 10 m/min in still water, wants to swim across river in shortest time. He should swim in direction a)due north. b)30 east of north. c) 30 north of west. d) 60 east of north. Justify you answer.

Grade:11

9 Answers

Ramesh V
70 Points
12 years ago

 

 

-------------> V river = 5m/min     

V man = 10 m/minsuppose width of river is x mts

Case 1

if he travels due north

then his resultant velocty ( due to velocity of downstream of water) acts 30 north of east and the distane to be covered to reach the other bank increses to = x/cos 30

i.e., 2x /31/2 and resultant velocity is 11.2 m/min

so time taken is x / 9.7 mins

Case 2

if he travels due 30 north of east

then his resultant velocty ( due to velocity of downstream of water) actsdue north and the distane to be covered to reach the other bank will be x mts and resultant velocity is 8.7 m/min

so time taken is x / 8.7 mins

So case 1 takes least time

he should travel due north(a)

 

 

AskIITianExpert Srijan.iitd
8 Points
12 years ago

he should swim due north!!!

his velocity in moving water is equal to his own velocity plus the velocity of the river.

now to cross the river in smallest possible time he should maximise the velocity in the north direction.

suppose he move at an angle of @ from the bank then the component due north is 10sin@m/min. because river velocity is zero in the north direction.

now this velocity due noth will be maximum when @ tends to 90 degrees.

hence he should move due north that is 90 degrees to the bank.

Aryan Malik
23 Points
4 years ago
Thus ,the resultant of the Velocity of man in still water (Vm)(=OA)and the Velocity of water (Vw)(=OB) is along OP, and is given by (V)(=OC)In right triangle OCA , we haveSin x= AC/OA=Vw/Vm =5/10 = 1/2x=30°Therefore, the man should start swimming at an angle of 30° (west of north)
Samuel Garry
61 Points
3 years ago
Yes we should move due north to cover the distance in the shortest time as then the velocity of river and man will get added up and distance would be covered in the shortest time
Prince Kumar
15 Points
2 years ago
If the man crossed the river with a angle @ to examined the river velocity then he should travel due north with velocity vcos@.so time taken to cross the river is d/vcos@.after differentiate both the sides of eqn we get @=0. So he should travel in north direction. 
Ajeet Tiwari
askIITians Faculty 86 Points
one year ago
hello student

The given condition can be shown as infigure
Time taken to cross the river,t=d/Vscosθ
for time to be minimum,cosθ=maximum
⇒θ=0
So, summer should swim due North.
642-881_mechanics.png


hope it helps
thankyou
Ajeet Tiwari
askIITians Faculty 86 Points
one year ago
hello students
642-1044_relative velocity.png

Hope it helps
Thankyou
MANAN
13 Points
one year ago
 
 

 

 

-------------> V river = 5m/min     

V man = 10 m/minsuppose width of river is x mts

Case 1

if he travels due north

then his resultant velocty ( due to velocity of downstream of water) acts 30 north of east and the distane to be covered to reach the other bank increses to = x/cos 30

i.e., 2x /31/2 and resultant velocity is 11.2 m/min

so time taken is x / 9.7 mins

Case 2

if he travels due 30 north of east

then his resultant velocty ( due to velocity of downstream of water) actsdue north and the distane to be covered to reach the other bank will be x mts and resultant velocity is 8.7 m/min

so time taken is x / 8.7 mins

So case 1 takes least time

HENCE HE SHOULD TRAVEL TOWARDS NORTH.

SO A OPTION IS CORRECT

Yash Chourasiya
askIITians Faculty 256 Points
one year ago
Dear Student

In order to swim across the river in the shortest time, the man should swim straight due north.
Because the velocity of the river is west to east and there is no component in north-south. So, it will not affect the man's time in order to swim to the other bank.

I hope this answer will help you.
Thanks & Regards
Yash Chourasiya

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