A plane is flying in a storm. A GPS system
measures the plane heading at an angle 23◦ west
of south, with a speed of 180 mph, relative to
the ground. A nearby weather station measures
the wind’s velocity to be 30◦ north of east, at a
speed of 50 mph.
Below, write a vector equation which gives
the plane’s velocity relative to the air (~vP/A)
in terms of its velocity relative to the ground
(~vP/G) and the air’s velocity relative to the
ground (~vA/G). Draw the corresponding vector
addition (or subtraction) to the right. What is
the magnitude and direction of the plane’s velocity
All angles are measured CW from +y-axis.
Vpa = 180[203o] + 50[60o].
Vpa = x+yi = (180*sin203+50*sin60) + (180*Cos203+50Cos60)i.
Vpa = -27 - 140.7i = 143mph[11o] W. 0f S. = 143mph[191o] CW.
TanA = x/y.
To find the plane's velocity relative to the air (~vP/A), we need to subtract the air's velocity relative to the ground (~vA/G) from the plane's velocity relative to the ground (~vP/G).
Let's break down the given information:
- The plane's velocity relative to the ground is given as 180 mph at an angle of 23° west of south.
- The air's velocity relative to the ground is given as 50 mph at an angle of 30° north of east.
To write the vector equation, we need to convert the given information into vector form. We'll use the following notation:
- Velocity relative to the ground (~vP/G) = vP/G
- Velocity relative to the air (~vP/A) = vP/A
- Velocity of the air relative to the ground (~vA/G) = vA/G
Given:
vP/G = 180 mph at an angle of 23° west of south
vA/G = 50 mph at an angle of 30° north of east
Expressing these velocities in vector form using magnitude and direction:
vP/G = 180 mph (23° west of south)
= 180 mph (270° - 23°) [Converting west of south to an angle]
= 180 mph (247°)
vA/G = 50 mph (30° north of east)
Now, we can write the vector equation using the principle of vector addition:
vP/A = vP/G - vA/G
Substituting the values:
vP/A = 180 mph (247°) - 50 mph (30°)
Now, to find the magnitude and direction of the plane's velocity relative to the air (~vP/A), we can calculate the resultant vector using trigonometry.
Magnitude of vP/A = sqrt[(180^2) + (50^2) - 2(180)(50)cos(217°)] [Applying the Law of Cosines]
Direction of vP/A = arctan[(180sin(247°) - 50sin(30°)) / (180cos(247°) - 50cos(30°))] [Applying the Law of Sines]
By substituting the values and performing the calculations, you can find the magnitude and direction of the plane's velocity relative to the air.