Sorry, my computer cannot upload the image. But I will try to give the solution in full details.
The resultant R
of two vectors P
( the angle between the two vectors P
) makes an angle
, such that,
= Q sin
/ ( P + Q cos
Here, we shall apply the above formula.
velocity of the man = v in the N.E direction.
velocity of the wind = w
the relative velocity of the man with respect to the wind = v - w = v + ( - w )
To the man, the wind appears to blow from the north i.e the direction of the resultant vector of v and - w (i.e the relative velocity of the man with respect to the wind ) is in the south direction.
Let the angle between the velocity of the man v
and the vector - w
Now, it is clear so far that since the vector v is in N.E direction and the vector sum of v and - w is the south direction then - w must be somewhere in the S-W direction.
So, now, framing the first equation,
tan 135 = w sin
/ ( v + w cos
( since, here the angle between the vector v and the resultant of v & ( - w ) is 135 degrees )
i.e - 1 = sin
/ ( f + cos
) where, f =
v / w.........eqn. (1)
Now when the speed of the man gets doubled the resultant velocity ( i.e the relative velocity of the man with respect to the wind ) shifts towards the vector whose magnitude has increased i.e towards the velocity of the man and now the resultant makes an angle of tan-12 east of north i.e the angle between the resultant and the 2v vector is (tan-12 - 45) degrees.
Now, we know, tan(x-y) = (tanx - tany) / (1+tanx.tany)
So, tan (tan-12 - 45) = ( tan tan-12 - tan 45 )/( 1 +tan-12 . tan 45 )
= (2-1) / (1+2.1) = 1 / 3
Now, let us form the second equation,
1/3 = w sin
/ ( 2 v + w cos
) = sin
( 2 f + cos
Those of you who wonder why I have taking f = v / w, then let me tell you in the above two equations we have three unknows v, w and
, so I've converted the two unknowns v,w into one unknown f = v / w. Now we have two unknows f &
and just two equations.
Eliminating f from the above two equations, we have,
= 5 sin
= - 1 / 5
Now, as estimated earlier that reverse velocity of the wind is in the S-W direction & the velocity of the man is in the N.E direction, so it is obvious that ,
= 180 + tan -1
( - 1 / 5 ) degrees = 180 - tan -1
Now, so the reverse velocity of the wind is ( 180 - tan -1 (1/5) - 135 ) degrees west of south, i.e [ 45 - tan -1 (1/5) ] degrees west of south.
The direction of the vector - w is [ 45 - tan -1 (1/5) ] degrees west of south.
So, the direction of vector w is [ 45 - tan -1 (1/5) ] degrees east of north.
Answer: The direction of the wind is [ 45 - tan -1 (1/5) ] degrees east of north.