A river d km wide is flowing at the rate straight v subscript 1 straight m divided by straight s. A swimmer which can swim with velocity straight v subscript 2 relative to water wants to cross the river straight across. Find how long will he take to cross the river?


As shown in figure, the path AB is the shortest path to cross the river. To cross the river straight across along path AB, the resultant velocity of swimmer and the water flow should be along AB i.e. perpendicular to the direction of water flow.

As shown in figure, the path AB is the shortest path to cross the riv
To meet this condition, let the swimmer start swimming upstream at angle 0 with AB. Therefore
tan 90 degree space equals space fraction numerator straight v subscript 2 sin left parenthesis 90 degree plus straight theta right parenthesis over denominator straight v subscript 1 plus straight v subscript 2 cos left parenthesis 90 degree plus straight theta right parenthesis end fraction
or space space space space space space space straight v subscript 1 plus straight v subscript 2 space cos left parenthesis 90 degree plus straight theta right parenthesis space equals space 0
or space space space space space space space space space straight v subscript 1 minus straight v subscript 2 space sinθ space equals space 0
or space space space space space space space space space space space space space space space space space space space space sinθ space equals space straight v subscript 1 over straight v subscript 2
Now   
         straight v equals straight v subscript 2 space cosθ space equals space straight v subscript 2 square root of 1 minus sin squared straight theta end root
space space space equals space straight v subscript 2 square root of 1 minus straight v subscript 1 squared over straight v subscript 2 squared end root space equals space square root of straight v subscript 2 squared minus straight v subscript 1 squared end root
Time taken by the swimmer to cross the river is,
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#6 {main}</pre>

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If a particle is in equilibrium under the action of three forces, then show that ratio of each force and sine of angle between the other two forces is constant?


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#6 {main}</pre> acting at point O be in equilibrium.

Let the three forces  acting at point O be in equilibrium.

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Two men A and B can swim in still water, at the rate of 5km/hr. The man A swims upstream a distance half km equal to width of river and B swims straight across the river and back again. Assuming that the velocity of river is 3 km/hr, find the difference in the time taken by them.

Here, velocity of man is,   v = 5 km/hr
Velocity of river is, u = 3 km/hr along QP

Here, velocity of man is,   v = 5 km/hrVelocity of river is, u = 3 k
Swimmer A:
Velocity of swimmer A w.r.t. bank while going from P to Q = v - u
  Velocity of swimmer A w.r.t. bank while coming back from Q to P = v + u
         Time taken by swimmer A to make the round trip is,
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#6 {main}</pre>
or                space space space straight t subscript 1 equals fraction numerator 2 cross times 5 cross times begin display style 1 half end style over denominator 5 squared minus 3 squared end fraction equals 5 over 16 hr space equals space 18 space min space 45 sec

Swimmer B:
To cross the river straight across man will dive at angle ***** (shown in figure) given by
      sinθ space equals straight u over straight v equals 3 over 5

Here, velocity of man is,   v = 5 km/hrVelocity of river is, u = 3 k
Therefore time taken by man B to make the round trip is,
                 straight t subscript 2 equals 2 cross times fraction numerator straight d over denominator straight v space cosθ end fraction equals 2 cross times fraction numerator 1 divided by 2 over denominator 5 cross times 0.8 end fraction equals 1 fourth hr equals 15 min
   The difference of the time taken by two swimmers is
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#6 {main}</pre>

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A stream of width d is flowing with velocity v. A person who can swim with speed V relative to water wishes to row straight across it. In what direction should he strike out and how long would it take him to cross the stream?


To row straight across the stream, the person must strike out at angle θ upstream, such that the resultant R of his own velocity and velocity of stream is perpendicular to velocity of stream.

∴     

        

                    

                            

∴                          

The swimmer should dive at an angle  with normal to river and upstream. 

Now the resultant velocity of swimmer and the stream is, 

                

               

            

Therefore the time taken by swimmer to cross the river is,

                 

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Point C is situated on the circumference of a circle of diameter AB. Show that triangle ACB is right angled triangle.


Let a be the radius of circle.

Let a be the radius of circle.

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