A small airplane has a speech of 200km/h with respect to the air. There is strong wind blowing at 77km/h at 23 north of east with respect to the Earth . A) in which direction should the plane head in order to land at an airport due north of its present location?B) what would be the plane's speech with respect to the ground ?
A) To determine the direction in which the plane should head to land at an airport due north, we need to find the resultant velocity of the airplane.
Step 1: We need to resolve the velocity of the wind into its north and east components.
Given:
Velocity of the wind = 77 km/h
Direction of the wind = 23° north of east
To find the north and east components of the wind, we can use trigonometry.
North Component of Wind = Velocity of Wind * cos(23°)
East Component of Wind = Velocity of Wind * sin(23°)
North Component of Wind = 77 km/h * cos(23°)
East Component of Wind = 77 km/h * sin(23°)
Step 2: To land at an airport due north, the plane's heading should be the opposite direction of the wind's east component. So, we need to subtract the east component of the wind from the plane's ground speed vector.
Given:
Ground Speed of the plane = 200 km/h
Plane's Heading = Ground Speed of Plane - East Component of Wind
Plane's Heading = 200 km/h - (77 km/h * sin(23°))
B) To determine the plane's speed with respect to the ground, we need to find the magnitude of the resultant velocity vector.
Step 1: To find the north and east components of the resultant velocity, we need to add the north component of the plane's velocity to the north component of the wind and the east component of the plane's velocity to the east component of the wind.
Given:
North Component of Plane's Velocity = Ground Speed of Plane * sin(Plane's Heading)
East Component of Plane's Velocity = Ground Speed of Plane * cos(Plane's Heading)
North Component of Resultant Velocity = North Component of Plane's Velocity + North Component of Wind
East Component of Resultant Velocity = East Component of Plane's Velocity + East Component of Wind
North Component of Resultant Velocity = (200 km/h * sin(Plane's Heading)) + (77 km/h * cos(23°))
East Component of Resultant Velocity = (200 km/h * cos(Plane's Heading)) + (77 km/h * sin(23°))
Step 2: The magnitude of the resultant velocity is given by the square root of the sum of the squares of the north and east components of the resultant velocity.
Magnitude of Resultant Velocity = sqrt((North Component of Resultant Velocity)^2 + (East Component of Resultant Velocity)^2)
Magnitude of Resultant Velocity = sqrt(((200 km/h * sin(Plane's Heading)) + (77 km/h * cos(23°)))^2 + ((200 km/h * cos(Plane's Heading)) + (77 km/h * sin(23°)))^2)
Note: The exact values of the sine and cosine functions may depend on the angle of the plane's heading.
To determine the direction in which the plane should head in order to land due north of its present location, we need to consider the vector addition of the velocity of the plane and the velocity of the wind.
A) To solve for the direction, we can use vector addition.
Step 1: Assign a coordinate system to the problem. Let's assume that east is the positive x-axis, and north is the positive y-axis.
Step 2: Determine the components of the velocities. Given the wind velocity of 77 km/h at 23° north of east and the plane's airspeed of 200 km/h, we can calculate the x and y components.
The x-component of the wind velocity is: 77 km/h * cos(23°)
The y-component of the wind velocity is: 77 km/h * sin(23°)
Step 3: Add the x-components and y-components separately to get the resultant vectors.
The x-component of the resultant vector (plane's airspeed + wind x-component):
= 200 km/h + (77 km/h * cos(23°))
The y-component of the resultant vector (wind y-component):
= (77 km/h * sin(23°))
Step 4: Find the direction of the resultant vector using the inverse tangent function.
The angle θ = arctan(resultant y-component / resultant x-component)
= arctan(wind y-component / (plane's airspeed + wind x-component))
Step 5: Convert the angle from radians to degrees to obtain the direction.
The direction in which the plane should head is the angle measured from the positive x-axis, so the angle obtained in radians should be converted to degrees.
B) The plane's groundspeed can be calculated by finding the magnitude of the resultant vector obtained in Step 3 using the Pythagorean theorem.
The magnitude of the resultant vector = sqrt((resultant x-component)^2 + (resultant y-component)^2)
Therefore, the plane's groundspeed is the square root of the sum of the square of the resultant x-component and the square of the resultant y-component.
Now that we've explained the steps, let's calculate the answers.
Calculations for A) and B) can be done using a scientific calculator or programming language.