A plane is flying 126 m/s east relative to the air. The wind is blowing 32 m/s south relative to the ground. Determine the plane's velocity relative to the ground.

Velocity (relative to the ground) magnitude =
Velocity (relative to the ground) direction = ° E of S

To determine the velocity of the plane relative to the ground, we need to find the resultant of the plane's velocity relative to the air and the wind's velocity relative to the ground.

Step 1: Convert the given velocities into vector form.
The plane's velocity relative to the air can be represented as 126 m/s in the east direction, which can be written as (126i), where i is the unit vector in the east direction.
The wind's velocity relative to the ground can be represented as 32 m/s in the south direction, which can be written as (-32j), where j is the unit vector in the south direction.

Step 2: Calculate the resultant vector.
To find the resultant vector, we add the vectors together, taking into account their magnitude and direction. Since the plane's velocity is relative to the air and the wind's velocity is relative to the ground, we need to consider their respective reference frames. We add the east and south components separately.

Plane's velocity (east component) + Wind's velocity (south component) = Resultant velocity

(126i) + (-32j) = Resultant velocity

Step 3: Calculate the magnitude and direction of the resultant vector.
To find the magnitude of the resultant vector, we use the Pythagorean theorem:

Magnitude of the resultant velocity = √[(magnitude of east component)^2 + (magnitude of south component)^2]
= √[(126^2) + (-32^2)]
≈ √(15876 + 1024)
≈ √16800
≈ 129.73 m/s

To find the direction of the resultant vector, we can use trigonometry. We can find the angle between the resultant vector and the south direction:

Direction of the resultant velocity = atan[(magnitude of east component) / (magnitude of south component)]
= atan(126 / 32)
≈ atan(3.94)
≈ 75.97°

Therefore, the plane's velocity relative to the ground is approximately 129.73 m/s in a direction of 75.97° east of south.