A person is travelling in the east direction with a speed of 2 km/hr with respect to the ground. he finds that the wind seems to blow from north. When he doubles his speed, the wind appears to come from north-east direction. determine the velocity of wind.

To determine the velocity of the wind, we can consider the vectors involved in the situation.

Let's call the velocity of the person relative to the ground (or the ground speed) as v_p and the velocity of the wind as v_w.

Given:
- The person is traveling in the east direction with a speed of 2 km/hr relative to the ground.
- The apparent direction of the wind is from the north when the person is traveling at v_p = 2 km/hr.
- When the person doubles their speed, the apparent direction of the wind is from the northeast.

To simplify the problem, let's assume the wind blows directly either from the north or the northeast (since the question mentions "appears to come from" those directions).

First, let's consider the situation where the person is traveling at v_p = 2 km/hr.
The apparent direction of the wind is from the north. This means that the wind vector (v_w) is perpendicular to the ground speed vector (v_p).

Now, when the person doubles their speed, the apparent direction of the wind is from the northeast. This means that the wind vector (v_w) lies between the north and east directions and forms an angle with respect to the ground speed vector (v_p).

To find the velocity of the wind, we need to find the vector sum of the ground speed (v_p) and the wind velocity (v_w) when the person is traveling at double the speed.

Let's denote the magnitude of the wind velocity as w and the angle it forms with respect to the ground speed vector as θ.

In the case where the person is traveling at v_p = 2 km/hr, we have:
v_p = 2 km/hr (due east)
v_w = w km/hr (due north)
v_p ⊥ v_w

Now, when the person is traveling at double the speed,
v_p = 4 km/hr (still due east)

The wind vector v_w forms an angle θ with respect to v_p and points towards the northeast direction.

We can use the concept of vector addition to find the wind velocity.

The x-component of the wind vector, v_wx, can be found using trigonometry:
v_wx = w * cos(θ)

The y-component of the wind vector, v_wy, can also be found using trigonometry:
v_wy = w * sin(θ)

Since the wind is coming from the northeast direction, both v_wx and v_wy should be positive.

Now, since the wind vector v_w is perpendicular to v_p when the person is traveling at v_p = 2 km/hr, we can conclude that v_wx = 0.

If we substitute v_wx = 0 into the equation v_wx = w * cos(θ), we get:
0 = w * cos(θ)

This implies that cos(θ) = 0, which means θ = 90 degrees.

Given that θ = 90 degrees, we can find the y-component of the wind vector using v_wy = w * sin(θ):
v_wy = w * sin(90) = w

So, the magnitude of the wind velocity is equal to the y-component of the wind vector, v_wy, which is equal to w.

Therefore, the velocity of the wind is w km/hr.

From the problem statement, we are given that the apparent direction of the wind is to the northeast when the person is traveling at double the speed. This implies that the x-component of the wind vector, v_wx, is positive.

Thus, from the given information, we can conclude that the wind is blowing towards the northeast direction at a speed of w km/hr.