A plane flies at an airspeed of 242 km/h. A wind is blowing at 72 km/h toward the direction 60° east of north.
What is the speed of the plane relative to the ground?
It depends on the heading of the airplane.
it is going directly north
To find the speed of the plane relative to the ground, we can use vector addition.
Let's represent the airspeed of the plane as vector A, which has a magnitude of 242 km/h.
Next, let's represent the wind speed as vector B, which has a magnitude of 72 km/h and is blowing in the direction 60° east of north.
Now, we need to break down vector B into its x and y components.
The y-component of vector B is B * sin(60°) = 72 km/h * sin(60°) = 62.4 km/h north.
The x-component of vector B is B * cos(60°) = 72 km/h * cos(60°) = 36 km/h east.
Since vector A is in the same direction as the plane's path, its x and y components are the same as the vector itself.
Now, we can add the x-components and y-components separately to find the resultant vector.
The x-component of the resultant vector is 242 km/h + 36 km/h = 278 km/h east.
The y-component of the resultant vector is 62.4 km/h + 0 km/h = 62.4 km/h north.
Using the Pythagorean theorem, we can find the magnitude of the resultant vector:
Magnitude = sqrt((278 km/h)^2 + (62.4 km/h)^2) ≈ 290.4 km/h.
Therefore, the speed of the plane relative to the ground is approximately 290.4 km/h.
To find the speed of the plane relative to the ground, we need to use vector addition.
First, we need to break down the velocities into their horizontal and vertical components.
The airspeed of the plane is given as 242 km/h. Since no direction is specified, we can assume it is along the positive x-axis (east). Therefore, the horizontal component of the airspeed is 242 km/h, and the vertical component is 0 km/h.
The wind is blowing at a speed of 72 km/h in a direction 60° east of north. To find the horizontal and vertical components of the wind velocity, we need to use trigonometry.
The horizontal component of the wind velocity is given by:
Horizontal component = Wind velocity * cos(angle)
Horizontal component = 72 km/h * cos(60°)
The vertical component of the wind velocity is given by:
Vertical component = Wind velocity * sin(angle)
Vertical component = 72 km/h * sin(60°)
Now, we can find the total horizontal and vertical components of the plane's velocity by adding the respective components of the airspeed and wind velocity.
Horizontal component of plane's velocity = Horizontal component of airspeed + Horizontal component of wind velocity
Vertical component of plane's velocity = Vertical component of airspeed + Vertical component of wind velocity
Finally, we can calculate the magnitude of the plane's velocity relative to the ground using the formula:
Magnitude of plane's velocity = sqrt((horizontal component)^2 + (vertical component)^2)
Let's calculate this step by step:
Horizontal component of wind velocity = 72 km/h * cos(60°) ≈ 72 km/h * 0.5 ≈ 36 km/h
Vertical component of wind velocity = 72 km/h * sin(60°) ≈ 72 km/h * 0.866 ≈ 62.352 km/h
Horizontal component of plane's velocity = 242 km/h + 36 km/h = 278 km/h
Vertical component of plane's velocity = 0 km/h + 62.352 km/h = 62.352 km/h
Magnitude of plane's velocity = sqrt((278 km/h)^2 + (62.352 km/h)^2) ≈ sqrt(77284 km^2/h^2 + 3880.706 km^2/h^2) ≈ sqrt(81164.706 km^2/h^2) ≈ 284.839 km/h
Therefore, the speed of the plane relative to the ground is approximately 284.839 km/h.