An airplane has a velocity relative to the air of 192 m/s in the westerly direction. If the wind has a speed relative to the ground of 59 m/s directed to the north, what is the speed of the plane relative to the ground?

It is the magnitude of the vector sum of the plane velocity relative to the air, and the wind velocity.

Since the two velocities you are adding are perpendicular, you can use the pythagorean theorem for the magnitude of the resultant.

sqrt[(192)^2 + (59)^2] = ___ m/s

200.86

To find the speed of the plane relative to the ground, we can use vector addition.

Let's break down the velocity of the plane into its components:

- Velocity of the plane in the x-direction (west/east): 192 m/s in the negative x-direction (due west).
- Velocity of the plane in the y-direction (north/south): 0 m/s (since the plane is not moving in this direction).

Now, let's break down the velocity of the wind into its components:

- Velocity of the wind in the x-direction (west/east): 0 m/s (since the wind is not blowing in this direction).
- Velocity of the wind in the y-direction (north/south): 59 m/s in the positive y-direction (due north).

To find the speed of the plane relative to the ground, we can add the velocity of the plane and the velocity of the wind using vector addition.

Speed of the plane relative to the ground = √[(velocity of the plane in the x-direction + velocity of the wind in the x-direction)^2 + (velocity of the plane in the y-direction + velocity of the wind in the y-direction)^2]

= √[(-192 m/s + 0 m/s)^2 + (0 m/s + 59 m/s)^2]

= √[(-192)^2 + 59^2]

= √[36992 + 3481]

= √40473

≈ 201.16 m/s

Therefore, the speed of the plane relative to the ground is approximately 201.16 m/s.

To find the speed of the plane relative to the ground, we can use vector addition. Let's break down the velocity of the plane relative to the ground into its horizontal and vertical components.

Given:
Velocity of the plane relative to the air (Vair) = 192 m/s (in the westerly direction)
Velocity of the wind relative to the ground (Vwind) = 59 m/s (directed to the north)

Step 1: Finding horizontal component
The horizontal component of the plane's velocity relative to the ground will be the same as its velocity relative to the air since the wind is blowing perpendicular to the plane's direction of motion.

Horizontal component (Vx) = Vair = 192 m/s

Step 2: Finding vertical component
The vertical component of the plane's velocity relative to the ground will be the addition of the vertical components of its velocity relative to the air and the wind's velocity relative to the ground.

Vertical component (Vy) = Vair + Vwind
= 192 m/s + 59 m/s
= 251 m/s

Step 3: Finding the resultant velocity
The speed of the plane relative to the ground (Vground) can be found using the Pythagorean theorem.

Vground = √(Vx² + Vy²)
= √(192² + 251²)
= √(36864 + 63001)
= √99865
≈ 315.83 m/s

Therefore, the speed of the plane relative to the ground is approximately 315.83 m/s.