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A beautiful girl is going eastwards with a velocity of $4\,Km/hr.$ The wind appears to blow directly from the north. She doubles her speed and the wind seems to come from the north east. The actual velocity of the wind is?
(A) $4\sqrt 2 {\text{ km/hr towards south east}}$
(B) $4\sqrt 2 {\text{ km/hr towards north west}}$
(C) $2\sqrt 2 {\text{ km/hr towards south east}}$
(D) ${\text{None of the above}}$

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Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: The direction changes if we double the speed. With that we can find the actual direction of wind when the girl is at rest position. We know the direction of wind at initial velocity $4\,Km/hr$ and the direction of wind after doubling the velocity. Draw a diagram representing the vectors of velocity and direction of the wind. From that we can understand the direction of the wind at rest position. Use Pythagorean Theorem to find the velocity of the wind at rest position.

Complete step by step answer
Velocity: The velocity of an object is defined as the change of rate of its position with respect to time and direction. Velocity is a vector quantity. It has both magnitude and direction i.e. the velocity is affected by the direction.
Speed: Speed describes how fast an object moves. Speed is defined as the magnitude of change of distance. Speed is a scalar quantity. It has only magnitude.
Given that, a beautiful girl is going eastwards with a velocity of $4\,Km/hr$ the wind appears to blow directly from the north.
The wind appears to blow directly from the north because the girl is moving with a velocity of $4\,Km/hr$
The wind appears to flow from the North east when she doubles the speed.
The initial velocity with which she travel when the wind appears to blow from north is $4\,Km/hr.$
When she doubles the speed the velocity will be two times the initial velocity.
Which means,
The velocity with which she travel when the wind appears to blow from north east $ = 2 \times 4\,Km/hr.$
         $ \Rightarrow 8\,Km/hr$
The wind appears to blow directly from the north when the girl is moving with a velocity of $4\,Km/hr.$ The wind appears to flow from the North east when she doubles the speed. Then the wind is supposed to blow from southeast initially when the girl is at rest position.
Let’s make it clear with the diagram.


AD is the actual direction of the wind when the girl is at rest position. BD is the direction of the wind when the girl moves with a velocity $4\,Km/hr$ and CD is the direction of the wind when she doubles the velocity$(8\,Km/hr)$.
From the diagram,
The direction of the wind when the girl is at rest position is southeast.
Pythagorean Theorem states that in a hypotenuse triangle the square of the hypotenuse is equal to the sum of squares of the other two sides.
Applying the Pythagorean Theorem to our triangle we get,
$ \Rightarrow {\text{A}}{{\text{D}}^2}{\text{ = }}A{B^2} + B{D^2}$
$ \Rightarrow {\text{AD = }}\sqrt {A{B^2} + B{D^2}} $
$ \Rightarrow \sqrt {{4^2} + {4^2}} $
$ \Rightarrow \sqrt {32} $
$ \Rightarrow 4\sqrt 2 $ towards Southeast

Hence the correct answer is option A) $4\sqrt 2 {\text{ km/hr towards south east}}$.

Note: Be careful when you draw the diagram, read the question thoroughly and then draw the diagram accordingly. Different questions may be asked using the same concept. If you draw an incorrect diagram then the whole answer will be wrong.