A pilot wants to fly from city A to city B, a distance of 394 km at an angle of 10.0 degrees west of north.
The pilot heads directly toward city B, with an air speed of 203 km/h.
After flying for 1.0 hours, the pilot finds she is 17.5 km off course to the west of where she expected to be, assuming there was no wind.
In what direction should the pilot have aimed her plane to fly directly to city B without being blown off course? Answer in degrees west of north.
using similar triangles, you can see that after flying the full 394 km, the plane would be
394/203 * 17.5 = 34 km west of city B.
So, in the diagram, let
P = place where the plane would end up after 394km
Draw a line east from P to intersect AB at Q, and the line north from A at R.
PR = 394 cosP
AR = 394 sinP
QR = AR tan10°
PR = 34+QR, so
34 + 394 sinP tan10° = 394 cosP
angle P = 75.1°
We want angle A in triangle APQ, which is how far off-course the plane was heading.
AR = 394 sinP = 381
cos(angle PAR) = 381/394, so angle PAR = 14.75°
But <PAR = <PAQ+10°, so <PAQ = 4.75°
So, the plane should have headed 5.25° west of north, rather than 10°.
To determine the direction the pilot should have aimed her plane to fly directly to city B without being blown off course, we need to consider the effect of the wind and calculate the required heading.
Let's break down the given information:
- The pilot wants to fly from city A to city B, a distance of 394 km.
- The desired heading (without wind) is an angle of 10.0 degrees west of north.
- The pilot's airspeed is 203 km/h.
- After flying for 1.0 hour, the pilot finds she is 17.5 km off course to the west of where she expected to be, assuming no wind.
To find the direction the pilot should have aimed her plane, we need to isolate the wind effect. We can determine the wind's velocity and direction by calculating the difference between the actual displacement and the expected displacement after accounting for the wind.
Let the wind velocity be represented by Vw in km/h and the wind direction as angle θw west of north. We want to find θw.
From the given information, we know that:
- The pilot flies for 1.0 hour at an airspeed of 203 km/h.
- The pilot expected to be 394 km away but is off course by 17.5 km to the west.
Considering the horizontal (north-south) displacement caused by the wind, we have:
Horizontal displacement = Vw * cos(θw) * time
We can calculate the wind velocity (Vw) using the given information:
Horizontal displacement = 17.5 km
Time = 1.0 hour
17.5 km = Vw * cos(θw) * 1.0 hour
Simplifying the equation, we can solve for Vw as:
Vw = 17.5 km / cos(θw) km
Now that we have the wind velocity, we can determine the direction the pilot should have aimed her plane. This is done by subtracting the wind's effect (magnitude and direction) from the desired heading.
Angle of desired heading - Angle of wind = Angle the pilot should have aimed her plane
Angle of desired heading = 10.0 degrees west of north
We now need to find the angle of the wind (θw). We have Vw = 17.5 km / cos(θw) km, which can be rearranged to:
cos(θw) = 17.5 km / Vw km
Taking the inverse cosine (cos^-1) of both sides gives us:
θw = cos^-1(17.5 km / Vw km)
Finally, we can calculate the direction the pilot should have aimed her plane by subtracting the angle of the wind (θw) from the desired heading:
Angle the pilot should have aimed her plane = 10.0 degrees west of north - θw
To obtain the numerical value, we need to substitute the wind velocity (Vw) calculated earlier into the equation. Please note that the actual numerical answer may vary depending on the value of Vw calculated.
Using the above steps, you can now calculate the direction the pilot should have aimed her plane to fly directly to city B without being blown off course.