An airplane's velocity with respect to the air is 580 miles per hour, and it is heading N 60 degrees W. The wind, at the altitude of the plane, is from the southwest and has a velocity of 60 miles per hour. What is the true direction of the plane, and what is its speed woth respect to the ground?
An airplane's velocity with respect to the air is 580 miles per hour, and it is heading 60 degrees Northwest. The wind, at the altitude of the plane, is from the southwest and has a velocity of 60 miles per hour. Draw a figure that gives a visual representation of the problem. What is the true direction of the plane, and what is its speed with respect to the ground?
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You are going to need graph paper, a protractor and a measuring ruler.
Draw an x-y Cartesian coordinate , x-y axis.
Problem starts at origin, (0,0). Call this point O
From the origin measure off a line 60 degrees from the y-axis.
The y-axis is North, (0 degrees) and your constructed line is the heading
of the airplane.
Using a suitable scale (I suggest 1 cm = 100 miles), measure off a
distance of 5.8 cm along your constructed line. At the 5'8 cm mark, call
this point A (for airplane)
Return to the origin. Measure off 45 degrees from north (the y-axis)
and draw another line. This is N 45 degrees W, and is really just the
line that represents the wind coming from the southwest, southwest
being 45 degrees to the west of south. Measure off 0.6 cm along this
line from the origin. Label that point W (for wind).
We now construct a parallelogram.
At point P, draw OW', parallel to original OW and the same length.
At point W, draw OP', parallel to original OP and the same length
Call the point of intersection of OW' and OP' point T (for true heading)
Draw the diagonal from O to T
OT is the "vector" that gives the true direction and speed of the
airplane.
Measure with your protractor the angle from the y-axis (it will be about
50 degrees, called N 50 degrees E) and the length in cm. of the diagonal OT. Multiply the measured cm. by 100 to get its speed
To find the true direction of the plane and its speed with respect to the ground, we need to use vector addition.
Step 1: Draw a diagram.
Draw a coordinate system on a piece of paper. Choose a scale for convenience, such as 1 cm = 100 mph. Mark the directions on the axis (North, South, East, West) and draw a line segment in the N 60 degrees W direction to represent the velocity of the plane with respect to the air (580 mph).
Step 2: Add the wind vector.
Draw a second line segment originating from the same point as the first one, representing the velocity of the wind (60 mph) in the southwest direction. Label this vector.
Step 3: Perform vector addition.
To find the resulting vector (true direction and speed with respect to the ground), add the two vectors using vector addition. This can be done graphically by placing the tail of the second vector at the head of the first vector and drawing the resultant vector from the tail of the first vector to the head of the second vector.
Step 4: Measure the resulting vector.
Measure the true direction angle from the North direction using a protractor. This angle will be the direction of the plane with respect to the ground. Measure the length of the resulting vector using a ruler or appropriate scale. This length will be the speed of the plane with respect to the ground.
Step 5: Calculate the true direction and speed.
Using the measured values, convert the true direction angle into a direction (in degrees) relative to North. For example, if the angle is measured as 45 degrees, the true direction would be N 45 degrees W. Additionally, multiply the measured length by the scale you chose to determine the actual speed of the plane with respect to the ground.
By following these steps, you should be able to determine the true direction and speed of the plane with respect to the ground.