A cessna plane flies with a speed of 150km/h relative to the air and points due east. A wind is blowing at a speed of 20km/h at 45 degrees south of east (ie the wind is blowing from the northwest). The resultant veloity of the plane relative to the ground is?

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

First, let's break down the given information:
- Cessna plane's velocity relative to the air = 150 km/h due east
- Wind's velocity = 20 km/h at 45 degrees south of east

To find the resultant velocity, we can use vector addition. We'll break down the given velocities into their respective eastward (x) and northward (y) components.

Plane's velocity components:
- Plane's velocity in the x-direction (eastward) = 150 km/h * cos(0°) = 150 km/h
- Plane's velocity in the y-direction (northward) = 150 km/h * sin(0°) = 0 km/h

Wind's velocity components:
- Wind's velocity in the x-direction (eastward) = 20 km/h * cos(-45°) ≈ 14.14 km/h
- Wind's velocity in the y-direction (northward) = 20 km/h * sin(-45°) ≈ -14.14 km/h

Now, we can add the x-components and y-components separately to find the resultant velocity.

Resultant velocity in the x-direction:
= Plane's velocity in the x-direction + Wind's velocity in the x-direction
= 150 km/h + 14.14 km/h
≈ 164.14 km/h eastward

Resultant velocity in the y-direction:
= Plane's velocity in the y-direction + Wind's velocity in the y-direction
= 0 km/h + (-14.14 km/h)
≈ -14.14 km/h (southward)

Finally, we can use the Pythagorean theorem to find the magnitude of the resultant velocity:

Resultant velocity = √[(Resultant velocity in the x-direction)² + (Resultant velocity in the y-direction)²]
= √[(164.14 km/h)² + (-14.14 km/h)²]
≈ √[26930 + 200]
≈ √27130
≈ 164.76 km/h

Therefore, the resultant velocity of the plane relative to the ground is approximately 164.76 km/h.

It is the vector sum of the wind velocity vector and the velocity of the plane with respect to the air.

You should be able to perform this operation yourself