An airplane flies at airspeed (relative to the
air) of 410 km/h . The pilot wishes to fly due
North (relative to the ground) but there is a
51 km/h wind blowing Southwest (direction
225◦
).
In what direction should the pilot head the
plane (measured clockwise from North)?
To determine the direction in which the pilot should head the plane, we need to calculate the resultant velocity.
Let's break down the velocities:
1. Airspeed (relative to the air) = 410 km/h
2. Wind speed and direction: 51 km/h in the southwest direction (225°).
To find the resultant velocity, we can use vector addition. We can break down the wind velocity into its northward and eastward components.
The northward component of the wind velocity can be calculated using trigonometry:
Northward component = Wind speed * sin(Wind direction)
= 51 km/h * sin(225°)
≈ -36.13 km/h (negative because it is southward)
The eastward component of the wind velocity can also be calculated using trigonometry:
Eastward component = Wind speed * cos(Wind direction)
= 51 km/h * cos(225°)
≈ -36.13 km/h (negative because it is westward)
Now, we can calculate the resultant velocity by adding the airspeed and the wind velocity components:
Northward component of resultant velocity = Airspeed * sin(0°) + Northward component of wind velocity
= 410 km/h * sin(0°) + (-36.13 km/h)
= -36.13 km/h
Eastward component of resultant velocity = Airspeed * cos(0°) + Eastward component of wind velocity
= 410 km/h * cos(0°) + (-36.13 km/h)
= 410 km/h - 36.13 km/h
= 373.87 km/h
Using these components, we can find the magnitude and direction of the resultant velocity.
Magnitude of resultant velocity = sqrt[(Northward component)^2 + (Eastward component)^2]
= sqrt[(-36.13 km/h)^2 + (373.87 km/h)^2]
≈ 378.36 km/h
Direction of resultant velocity = arctan[(Northward component)/(Eastward component)]
= arctan[(-36.13 km/h)/(373.87 km/h)]
≈ -5.50°
The pilot should head the plane in a direction approximately 5.50° west of north (measured clockwise from North) to counteract the effect of the wind and fly due north relative to the ground.
To determine the direction in which the pilot should head the plane, we need to consider the effect of the wind on the plane's actual ground speed (the speed relative to the ground).
Step 1: Break down the velocities into their North and East components:
The airspeed of the plane is 410 km/h, and the wind is blowing at 51 km/h in the 225° direction (Southwest).
The North component of the wind is given by:
North component of wind = 51 km/h * cos(225°)
The East component of the wind is given by:
East component of wind = 51 km/h * sin(225°)
Step 2: Calculate the total ground speed of the plane:
The ground speed of the plane is the vector sum of the airspeed and the wind speed.
North component of ground speed = airspeed + North component of wind
East component of ground speed = East component of wind
Step 3: Determine the direction in which the pilot should head the plane:
The direction in which the pilot should head the plane can be found using the tangent function:
Direction = arctan(East component of ground speed / North component of ground speed)
Step 4: Convert the direction from radians to degrees and adjust to find the direction clockwise from North:
Express the direction in radians and convert it to degrees. Then, adjust the direction by adding 180° to find the clockwise angle from North.
Therefore, by following these steps, we can find the direction in which the pilot should head the plane.