THE CLASSIC RIVER - BOAT/SWIMMER PROBLEM

THIS WAS OBVIOUS
U ADD 2 TIMES A NUMBER TO 21 THEN HALF IT AND SUBSTRACT THE ORIGINAL ,IT WILL GIVE YOU 21/2 = 10.5

u can study a book on higher algebra . it will really help u

the answer to this lies in number theorem that says that such equation donot have any soln for n greater than 2

not many shortcuts are available but u cud refer m.l.khanna for a good reference

ONE OF THE BEST BOOK THAT I HAVE READ ON ORGANIC CHEMISTRY OTHER THAN MORRISON AND BOYD.IT HAS DETAILED EXPLANATIONS OF ALL TYPES OF REACTIONS

presentation is quite nice

shud say ur quite smart

quite an article i must say

good job

I HAVE MADE AN ARTICLE FOR YOU . PLZ READ AND RATE IT IF IT IS HELPFUL

YOU CAN VISIT MY ARTICLE RELATED TO THIS

THE CLASSIC RIVER - BOAT/SWIMMER PROBLEM

MANY A TIMES THE APPLICATION OF VECTOR IS QUITE DIFFICULT FOR THOSE WHO HAVE LITTLE OR NO PRIOR KNOWLEDGE OF IT.SO HERE I HAVE TRIED TO BRING OUT A GENERAL EQUATION FOR THE DRIFT AND ANGLE THAT CAN BE USED BY THE READERS TO SOLVE

?CROSSING THE RIVER? TYPE OF QUESTION.

CONSIDER A GENERAL CASE WHEN THE SPEED OF A BOAT OR SWIMMER IS ?v? WITH RESPECT TO STILL WATER. ALSO THE SPEED OF THE STREAM IS ?u?. THE RIVER IS ?l? UNITS WIDE.SUPPOSE THE DRIFT FROM HIS ACTUAL PATH IS x.

THE VELOCITY OF OBJECT IN THAT DIRECTION IS =

STREAM SPEED + COMPONENT OF OBJECT SPEED IN THAT DIRECTION.

w = u + v*cos? where ? is the angle between the swimmer and the Stream velocity

HENCE THE DRIFT IN THAT DIRECTION IS

x = ( u + v*cos? )*t where t is the time taken to cross the river ??.(1)

NOTE : HERE IT IS IMPORTANT TO NOTE THAT IF THE OBJECT IS MOVING AGAINST THE STREAM ?>90. HENCE cos ? IS NEGETIVE . THIS REDUCES THE DRIFT AS IS EXPECTED IN REAL LIFE SITUATION.

l = v*t*sin ?

t = l/v*sin ? , NOW SUBSTITUTING THIS IN (1) WE HAVE

x = { (u/v*sin ?) + cot ? }

THESE VALUES ARE GENERALLY GIVEN IN THE QUESTION AND HENCE FINDING DRIFT NOW TAKES NO TIME.NOW IF NOW THE VALUE OF u/v OR IF IT CAN BE FOUND NUMERICALLY THEN LET u/v=n .

SO, x = { n*cosec ? + cot ? }*l

HENCE FOR MINIMUM DRIFT ( dx/d? ) = 0;

-n*cosec? ? cosec?*cosec? = 0

OR cosec? = 0 OR cos? = -1/n

BUT cosec? CANNOT BE ZEROAS IT IS NOT IN RANGE (-?,1]u[1, ?)

SO ? = cos?( -1/n ) .................(2)

HENCE THE ANGLE CAN BE FOUND OUT BY JUST KNOWING n.

CAUTION: THE SECOND EQUATION CAN BE USED ONLY WHEN u>v OR ELSE THE VALUE OF cos? WOULD BE OUT OF RANGE.

ALTHOUGH I THINK YOUR QUESTION HAS NOT BEEN SUBMITTED PROPERLY I?LL STILL TRY TO ANSWER IT AND YOU REPLY ME BACK AND TELL ME WEATHER I AM CORRECT.

The resultant of a , b is a+b;

Let the angle between a and b be 3? and since a+b trisects it

Let it make an angle ? with a and 2? with b

|a|=A and |b|=B

Using dot product btw a and a+b, we have

a.(a+b) = |a||a+b|cos?

a.a+a.b= A|a+b|cos?

A*A+ ABcos3? = |a+b| A cos?

Or A+ Bcos3? = |a+b|cos? ????(1)

Similarly using dot product btw b and a+b, we have

B + Acos3? = |a+b|cos2? ?????(2)

Dividing 1 and 2 and solving further we have

A(2sin3?sin?) = B(2sin3?sin2?)

Asin? = B(2sin?cos?)

cos? = A/2B

or ? = cos?(A/2B) i.e. angle btw a and a+b

now angle btw a and b is 3? i.e. 3{cos?(A/2B)}

now |a+b||a+b| = |a||a| + |b||b| + 2a.b

|a+b||a+b| = |a||a| + |b||b| + 2|a||b|cos3? (since a.b= |a||b|cos3?)

= A*A + B*B + A*B(4cos3?-3cos?)

Substituting value of ? and solving we have

|a+b|2 = {(a2-b2)/b}2

i.e |a+b| = {(a2-b2)/b}

well the idea is very simple.just as solving triangles and complex no.s can be used in 2-D to solve geometric problems such as proof of pytogorus theorem vectors can be used to do the same in 3-D. Advantage vector has is any point can be taken as origin (unlike 3-D geo) that makes problem solving lot easier

there a lot many other advantages that u can learn with practice and intution.

hope i was able to help you

two points A and B are given on the curve y=x² such that vector OA.i=1 and OB.i= -2 find |2OA-3OB| when i is unit vector and OA and OB are vectors directed from origin to A and B respectively^.

a very good for conceptual understanding of waves and a variety of extra quick shortcut methods

gr88 job hope to c such more

THE CLASSIC RIVER - BOAT/SWIMMER PROBLEM

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