An air craft is traveling due north with avelocity of 100 m/s, astrong wind blows from the west with avelocity of 25m/s find resultant velocity

Vr = 25 + 100i = 103m/s[76o].

Good

To find the resultant velocity of the aircraft, we can use vector addition.

Given:
Aircraft velocity (V_plane) = 100 m/s due north
Wind velocity (V_wind) = 25 m/s towards west

We can treat the north direction as the positive y-axis and the west direction as the negative x-axis.

The x-component of the resulting velocity (V_resultant_x) can be found by subtracting the wind velocity from the aircraft velocity:

V_resultant_x = V_plane_x - V_wind
= 0 - 25
= -25 m/s

The y-component of the resulting velocity (V_resultant_y) remains unchanged since the wind is blowing perpendicular to the direction of aircraft's motion.

V_resultant_y = V_plane_y
= 100 m/s

Now, we can use the Pythagorean theorem to find the magnitude of the resultant velocity (V_resultant) by combining the x and y components:

V_resultant = √(V_resultant_x^2 + V_resultant_y^2)
= √((-25)^2 + 100^2)
= √(625 + 10000)
= √10625
≈ 103.06 m/s

The direction of the resultant velocity can be found using trigonometry:

θ = tan^(-1)(V_resultant_y / V_resultant_x)
= tan^(-1)(100 / -25)
= tan^(-1)(-4)
≈ -75.96 degrees

Therefore, the resultant velocity of the aircraft is approximately 103.06 m/s at an angle of -75.96 degrees with respect to the north direction.

To find the resultant velocity, we need to use vector addition. The given velocities are in different directions, so we can use the Pythagorean theorem to calculate the magnitude of the resultant vector and trigonometry to determine its direction.

1. Draw a diagram: Draw a coordinate system with the positive x-axis representing east and the positive y-axis representing north. Since the wind blows from the west, its velocity will point in the negative x-direction. The aircraft's velocity is directly north, so it will have no x-component.

```
N
|
|
W ------|-----E
/
/
S
```

2. Calculate the x-component of the resultant velocity: Since the wind blows in the -x direction with a velocity of 25 m/s, the x-component of the resultant velocity is -25 m/s.

3. Calculate the y-component of the resultant velocity: The aircraft is traveling due north with a velocity of 100 m/s, so the y-component of the resultant velocity is +100 m/s.

4. Use the Pythagorean theorem to find the magnitude of the resultant velocity:
`Resultant Velocity = sqrt(x-component^2 + y-component^2)`

`Resultant Velocity = sqrt((-25)^2 + 100^2)`

`Resultant Velocity ≈ sqrt(625 + 10000) ≈ sqrt(10625) ≈ 103.06 m/s`

5. Use trigonometry to find the angle of the resultant velocity:
`tan(θ) = y-component / x-component`

`tan(θ) = 100 / -25`

`θ ≈ -75.96°` (Note: The negative sign indicates that the resultant vector is directed towards the south of west.)

Therefore, the resultant velocity is approximately 103.06 m/s directed at an angle of -75.96° with respect to the positive x-axis or westward.