A small aircraft, on a heading of 225 degrees, is cruising at 150 km/h. It is encountering a wind blowing from a bearing of 315 degrees at 35 km/h.

Determine the aircraft's ground velocity.

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To determine the aircraft's ground velocity, we need to find the resultant velocity vector by adding the aircraft's velocity vector to the wind's velocity vector.

Step 1: Convert the aircraft's heading and wind's bearing to Cartesian coordinates.

The aircraft's heading of 225 degrees can be converted to Cartesian coordinates as follows:
x = cos(225°) = -0.7071
y = sin(225°) = -0.7071

The wind's bearing of 315 degrees can be converted to Cartesian coordinates as follows:
x = cos(315°) = 0.7071
y = sin(315°) = -0.7071

Step 2: Calculate the aircraft's velocity vector (A) and the wind's velocity vector (W).

The aircraft's velocity vector (A) can be calculated by multiplying the aircraft's speed (150 km/h) with the aircraft's heading unit vector:
A = (150 km/h) * (-0.7071 i - 0.7071 j)

The wind's velocity vector (W) can be calculated by multiplying the wind's speed (35 km/h) with the wind's bearing unit vector:
W = (35 km/h) * (0.7071 i - 0.7071 j)

Step 3: Add the aircraft's velocity vector (A) and the wind's velocity vector (W) to find the resultant velocity vector (R).

R = A + W

Step 4: Determine the magnitude and direction of the resultant velocity vector (R) to find the aircraft's ground velocity.

The magnitude of the resultant vector (R) can be calculated using the Pythagorean theorem:
|R| = sqrt((Rx)^2 + (Ry)^2), where Rx and Ry are the x and y components of R, respectively.

The direction of the resultant vector (R) can be determined using the inverse tangent function:
θ = atan(Ry / Rx)

Now, let's calculate the aircraft's ground velocity:

R = (-0.7071 * 150 km/h + 0.7071 * 35 km/h) i + (- 0.7071 * 150 km/h - 0.7071 * 35 km/h) j
= (-106.07 + 24.8) i + (-106.07 - 24.8) j
= -81.27 i - 130.87 j

|R| = sqrt((-81.27)^2 + (-130.87)^2) = 154.28 km/h

θ = atan((-130.87) / (-81.27)) = 57.7 degrees (assuming the positive x-axis as 0 degrees)

Therefore, the aircraft's ground velocity is 154.28 km/h on a heading of 57.7 degrees.

To determine the aircraft's ground velocity, we need to consider both the aircraft's airspeed and the effect of the wind.

First, let's determine the aircraft's velocity relative to the still air. We can use vector addition to find the aircraft's velocity by breaking it down into its northward and eastward components.

Given that the aircraft is heading on a bearing of 225 degrees, we can represent the aircraft's velocity relative to the still air as V_aircraft = (V_north, V_east). The airspeed is given as 150 km/h.

To find the northward component, we can use trigonometry. Since the angle between the aircraft's heading and the north direction is 225 degrees, the northward component of the velocity can be found using sine.

V_north = V_aircraft * sin(225°)
V_north = 150 km/h * sin(225°)

Next, we find the eastward component of the velocity. We can use cosine since the angle is between the aircraft's heading and the east direction.

V_east = V_aircraft * cos(225°)
V_east = 150 km/h * cos(225°)

Now, let's determine the wind's velocity relative to the ground. Given that the wind is blowing from a bearing of 315 degrees at 35 km/h, we can represent the wind velocity as V_wind = (V_wind_north, V_wind_east).

To find the northward component of the wind velocity, we can use the same approach as before:

V_wind_north = V_wind * sin(315°)
V_wind_north = 35 km/h * sin(315°)

Similarly, we can find the eastward component of the wind velocity:

V_wind_east = V_wind * cos(315°)
V_wind_east = 35 km/h * cos(315°)

Now, let's determine the aircraft's ground velocity by adding the velocity of the aircraft relative to the still air (V_aircraft) to the velocity of the wind relative to the ground (V_wind).

V_ground_north = V_north + V_wind_north
V_ground_east = V_east + V_wind_east

Finally, we can calculate the magnitude and direction of the aircraft's ground velocity using the Pythagorean theorem and trigonometry.

V_ground = sqrt(V_ground_north^2 + V_ground_east^2)
Direction = arctan(V_ground_north / V_ground_east)

Plugging in the values we calculated above, you can determine the aircraft's ground velocity.