An airplane is flying at 550 km/h on a heading of 080 degrees. The wind is blowing at 60 km/h from a bearing of 120 degrees. Determine the ground velocity of the airplane.
I am having trouble drawing this diagram.
To solve this problem, it's important to visualize the given information correctly. Here's how you can draw the diagram and solve the problem step by step:
Step 1: Draw a horizontal line to represent the ground. This is your reference line.
Step 2: Mark a point on the reference line to represent the starting position of the plane. Label it, for instance, as "A."
Step 3: Draw an arrow pointing to the right from the starting position. Label it with the speed of the plane: 550 km/h.
Step 4: From point A, draw a line in the direction of 080 degrees. This line represents the heading of the plane. Label its endpoint as "B."
Step 5: At point B, draw a line that is 120 degrees clockwise from the heading line. This line represents the direction from which the wind is blowing. Label the endpoint of this line as "C."
Step 6: Measure and label the length of line segment BC as 60 km/h.
Step 7: From point B, draw a line perpendicular to the reference line, pointing downwards. This line represents the component of wind blowing in the same direction as the plane's heading (called the "headwind").
Step 8: From point C, draw a line perpendicular to the reference line, pointing upwards. This line represents the component of wind blowing against the direction of the plane's heading (called the "crosswind").
Step 9: Label the length of the headwind line as "D," and the length of the crosswind line as "E."
Step 10: Now, consider the headwind and the plane's velocity arrow (550 km/h):
- The headwind (D) and the plane's velocity arrow (550 km/h) form a right-angled triangle.
- The plane's velocity arrow is the hypotenuse of this triangle.
- The magnitude of the plane's velocity (ground speed) can be found using the Pythagorean theorem:
ground speed^2 = 550 km/h^2 + D^2
Step 11: Consider the crosswind and the plane's velocity arrow (550 km/h):
- The crosswind (E) and the plane's velocity arrow (550 km/h) also form a right-angled triangle.
- The crosswind (E) is one of the legs of this triangle.
Step 12: Use trigonometric ratios to find the magnitude of the crosswind (E). Since we know the angle and the hypotenuse of this triangle, we can use the sine ratio:
sin(angle) = E / 550 km/h
Step 13: Once you have calculated the magnitudes of the headwind (D) and the crosswind (E), you can determine the ground velocity using vector addition.
- The ground velocity of the plane can be determined by adding its velocity (550 km/h) to the headwind (D) and crosswind (E).
- The magnitude of the ground velocity is equal to the magnitude of the resultant vector obtained from adding these three vectors.
I hope this step-by-step explanation helps you in drawing the diagram and solving the problem.