A plane has an airspeed of 200 m/s due north and is subject to a westerly wind of 50 m/s. What is the planes speed relative to the ground?
To find the plane's speed relative to the ground, we can use vector addition. We'll need to break down the velocities into their horizontal and vertical components.
Given:
Airspeed of the plane (due north): 200 m/s
Wind speed (westerly): 50 m/s
Step 1: Break down the velocities into components
The airspeed has a vertical component of 200 m/s (northward) and no horizontal component.
The wind speed has a horizontal component of -50 m/s (westward) and no vertical component.
Step 2: Add the corresponding components
To find the net vertical component, we need to add the vertical components of both the airspeed and the wind speed:
Net vertical component = 200 m/s + 0 m/s = 200 m/s (northward)
To find the net horizontal component, we need to add the horizontal components of both the airspeed and the wind speed:
Net horizontal component = 0 m/s + (-50 m/s) = -50 m/s (westward)
Step 3: Use the Pythagorean theorem to find the magnitude of the resulting vector
The magnitude of the resulting vector is the square root of the sum of the squares of its components:
Magnitude of the resulting vector = √(vertical component^2 + horizontal component^2)
Magnitude of the resulting vector = √((200 m/s)^2 + (-50 m/s)^2)
Step 4: Calculate the magnitude
Magnitude of the resulting vector = √(40,000 m^2/s^2 + 2,500 m^2/s^2)
Magnitude of the resulting vector = √42,500 m^2/s^2
Step 5: Simplify the value
Magnitude of the resulting vector = √(42,500) m/s
Magnitude of the resulting vector ≈ 206.15 m/s
Therefore, the plane's speed relative to the ground is approximately 206.15 m/s.
To find the plane's speed relative to the ground, we need to consider the effect of the wind on the plane's motion.
We can use vector addition to solve this problem. Let's break down the velocities into their components. The plane's airspeed can be broken down into a northward component of 200 m/s and no eastward component since it is completely traveling north. The wind velocity can be broken down into a westward component of 50 m/s and no northward component.
Now, add the components of the airspeed and wind velocity. The northward component of the airspeed remains unaffected by the wind, so it will stay the same at 200 m/s. The westward component of the airspeed will be affected by the wind and reduce the groundspeed.
To find the resultant ground speed, we need to calculate the magnitude of the resultant velocity vector. This can be done using the Pythagorean theorem:
resultant magnitude = √(northward component^2 + westward component^2)
In this case, the northward component is 200 m/s and the westward component is -50 m/s (since it is in the opposite direction). Plugging these values into the formula:
resultant magnitude = √(200^2 + (-50)^2)
= √(40000 + 2500)
= √42500
≈ 206.2 m/s
So, the plane's speed relative to the ground is approximately 206.2 m/s.