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Physics-

General

Easy

Question

The wind appears to blow from the north to a man moving in the north-east direction. When he doubles his velocity the wind appears to move in the direction $cot to the power of negative 1 end exponent invisible function application left parenthesis 2 right parenthesis$ east of north. The actual direction of the wind is

$\sqrt{2} v$ towards east
$\frac{v}{\sqrt{2}}$ towards west
$\sqrt{2} v$ towards west
$\frac{v}{\sqrt{2}}$ towards east

The correct answer is: $fraction numerator v over denominator square root of 2 end fraction$ towards east

$stack v with rightwards arrow on top subscript m a n end subscript$ = $fraction numerator v over denominator square root of 2 end fraction stack i with hat on top plus fraction numerator v over denominator square root of 2 end fraction stack j with hat on top$
Let $stack v with rightwards arrow on top subscript w i n d end subscript equals a stack i with hat on top plus b stack j with hat on top$
$rightwards double arrow blank stack v with rightwards arrow on top subscript w i n d divided by m a n end subscript$ = $open parentheses a minus fraction numerator v over denominator square root of 2 end fraction close parentheses stack i with hat on top plus open parentheses b minus fraction numerator v over denominator square root of 2 end fraction close parentheses stack j with hat on top$
$rightwards double arrow blank tan invisible function application theta equals fraction numerator b minus fraction numerator v over denominator square root of 2 end fraction over denominator a minus fraction numerator v over denominator square root of 2 end fraction end fraction equals tan invisible function application 270 degree$
$rightwards double arrow blank a minus fraction numerator v over denominator square root of 2 end fraction equals 0$
$rightwards double arrow a equals fraction numerator v over denominator square root of 2 end fraction blank rightwards double arrow stack v with rightwards arrow on top subscript w i n d end subscript equals fraction numerator v over denominator square root of 2 end fraction stack i with hat on top plus b stack j with hat on top$
when the man doubles his speed
$stack v with rightwards arrow on top subscript m a n end subscript superscript ´ end superscript equals 2 open parentheses fraction numerator v over denominator square root of 2 end fraction stack i with hat on top plus fraction numerator v over denominator square root of 2 end fraction stack j with hat on top close parentheses equals square root of 2 blank end root left parenthesis v stack i with hat on top plus v stack j with hat on top right parenthesis$
$rightwards double arrow blank stack v with rightwards arrow on top ´ subscript w i n d divided by m a n end subscript$ = $open parentheses fraction numerator v over denominator square root of 2 end fraction minus square root of 2 blank end root v close parentheses stack i with hat on top plus left parenthesis b minus square root of 2 blank end root v right parenthesis stack j with hat on top$
$rightwards double arrow blank t a n theta to the power of ´ end exponent equals fraction numerator b minus square root of 2 blank end root over denominator fraction numerator v over denominator square root of 2 minus square root of 2 blank end root v end fraction end fraction equals fraction numerator 2 v minus square root of 2 blank b end root over denominator v end fraction$
But $theta to the power of ´ end exponent equals 270 degree minus cot to the power of negative 1 end exponent invisible function application open parentheses 2 close parentheses$
$rightwards double arrow$ tan [ $270 degree minus cot to the power of negative 1 end exponent invisible function application open parentheses 2 close parentheses$ ] $equals blank fraction numerator 2 v minus square root of 2 b end root over denominator v end fraction$
$rightwards double arrow$ cot[ $cot to the power of negative 1 end exponent invisible function application left parenthesis 2 right parenthesis right square bracket equals fraction numerator 2 v minus square root of 2 b end root over denominator v end fraction$
$rightwards double arrow 2 v equals 2 v minus square root of 2 blank b end root equals b equals 0$
$therefore blank stack v with rightwards arrow on top subscript w i n d end subscript$ = $fraction numerator v over denominator square root of 2 end fraction stack i with hat on top$

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