An airplane flies at airspeed (relative to the
air) of 470 km/h . The pilot wishes to fly due
North (relative to the ground) but there is a
64 km/h wind blowing Southwest (direction
225◦).
In what direction should the pilot head the
plane (measured clockwise from North)?
Answer in units of ◦.
To determine the direction the pilot should head the plane, we need to find the resultant velocity by considering the velocity of the plane and the velocity of the wind.
The velocity of the plane with respect to the ground is the vector sum of the airspeed and the wind velocity.
Let's break down the velocities into their components:
Given:
Airplane airspeed: 470 km/h
Wind speed: 64 km/h
Components of airplane airspeed:
North (N) component: 470 km/h * cos(0°)
East (E) component: 470 km/h * sin(0°)
Components of wind:
North (N) component: 64 km/h * cos(225°)
East (E) component: 64 km/h * sin(225°)
Now, we can sum up the components of the airspeed and wind to find the resultant velocity:
North (N) component of resultant velocity:
= (470 km/h * cos(0°)) + (64 km/h * cos(225°))
East (E) component of resultant velocity:
= (470 km/h * sin(0°)) + (64 km/h * sin(225°))
To find the direction the pilot should head, we can use the inverse tangent function:
θ = arctan((E component of resultant velocity) / (N component of resultant velocity))
Now, let's calculate the values:
North (N) component of resultant velocity:
= (470 km/h * cos(0°)) + (64 km/h * cos(225°))
= 470 km/h * 1 + (-64 km/h * 0.7071)
= 470 km/h - 45.49 km/h
= 424.51 km/h
East (E) component of resultant velocity:
= (470 km/h * sin(0°)) + (64 km/h * sin(225°))
= 470 km/h * 0 + (-64 km/h * 0.7071)
= 0 - 45.49 km/h
= -45.49 km/h
θ = arctan((-45.49 km/h) / (424.51 km/h))
θ = arctan(-0.1073)
Using a calculator, we find:
θ ≈ -6.12°
Therefore, the pilot should head the plane approximately 6.12° west of north (measured clockwise from north).
To find the direction the pilot should head the plane, we must consider both the airspeed of the plane and the wind speed and direction.
1. First, we need to find the resultant velocity of the airplane. This is the vector sum of the airspeed and the wind velocity.
To calculate the horizontal component of the resultant velocity:
Relative horizontal velocity = Airspeed * cos(225°)
To calculate the vertical component of the resultant velocity:
Relative vertical velocity = Airspeed * sin(225°)
2. Now we add the horizontal component of the wind velocity to the horizontal component of the resultant velocity, and the vertical component of the wind velocity to the vertical component of the resultant velocity to find the resultant velocity itself.
Horizontal component of resultant velocity = Relative horizontal velocity + Wind velocity * cos(225°)
Vertical component of resultant velocity = Relative vertical velocity + Wind velocity * sin(225°)
3. Finally, we can find the direction the pilot should head by calculating the angle between the resultant velocity vector and the North direction. This can be done using the arctan function:
Direction = arctan(vertical component of resultant velocity / horizontal component of resultant velocity)
Now we can substitute the given values: Airspeed = 470 km/h and Wind velocity = 64 km/h, and calculate the direction using the above steps.
Relative horizontal velocity = 470 km/h * cos(225°) = -333.63 km/h
Relative vertical velocity = 470 km/h * sin(225°) = -333.63 km/h
Horizontal component of resultant velocity = -333.63 km/h + 64 km/h * cos(225°) = -135.9 km/h
Vertical component of resultant velocity = -333.63 km/h + 64 km/h * sin(225°) = -467.9 km/h
Direction = arctan(-467.9 km/h / -135.9 km/h)
Calculating this angle will give us the direction the pilot should head the plane, measured clockwise from North.
Vp + 64[225] = 470[90o] = 470i.
Vp + 64*Cos225 + i64*sin225 = 470i.
Vp - 45.3 - 45.3i = 470i.
Vp = 45.3 + 515.3i = 517km/h[85o]. =
velocity of the plane.
Direction = 85o N. of E. = 5o E. of N.