Does this triangle have to be flipped when finding the magnitudes and angles?

• MHB
• Raerin
In summary, when solving relative velocity problems in calculus and vectors class, the direction of the plane may need to be flipped when drawing the triangle to find the magnitudes and angles. One way to solve this type of problem is by using the Law of Cosines and the other is by using vector addition.
Raerin
In those kind of relative velocity questions in calculus & vectors class, does the direction of the plane need to be flipped when drawing the triangle to find the magnitudes and angles?

Example of a question:
An airline pilot has her controls set to fly at an air speed of 615 km/h at an azimuth bearing of 40 degrees. A wind is blowing from an azimuth bearing of 205 degrees at 80 km/h. Determine the velocity of the plane relative to the ground.

For this question, I also have troubles finding the angles between vectors when I draw the triangle. A link to an illustration on Paint or some other drawing program would be really helpful!

Thanks :)

Here are two ways to work the problem:

a) Law of Cosines:

Consider the following diagram:

View attachment 1546

The angle between the plane's velocity vector and that of the wind is:

$$\displaystyle \theta=\left(180-((90-(205-180))-(90-40)) \right)^{\circ}=165^{\circ}$$

Hence:

$$\displaystyle R=\sqrt{615^2+80^2-2\cdot615\cdot80\cos\left(165^{\circ} \right)}\approx692.583642102$$

$$\displaystyle \vec{R}=\left\langle 615\cos\left(50^{\circ} \right)+80\cos\left(65^{\circ} \right),\sin\left(50^{\circ} \right)+80\sin\left(65^{\circ} \right) \right\rangle$$

$$\displaystyle R=\left|\vec{R} \right|=\sqrt{\left(615\cos\left(50^{\circ} \right)+80\cos\left(65^{\circ} \right) \right)^2+\left(\sin\left(50^{\circ} \right)+80\sin\left(65^{\circ} \right) \right)^2}\approx692.583642102$$

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1. What is "plane to ground velocity"?

"Plane to ground velocity" refers to the speed at which an aircraft is moving relative to the surface of the earth.

2. How is "plane to ground velocity" different from airspeed?

"Plane to ground velocity" takes into account the movement of the aircraft over the ground, while airspeed only measures the speed of the aircraft through the air.

3. How is "plane to ground velocity" calculated?

"Plane to ground velocity" is calculated by combining the aircraft's airspeed with the wind speed and direction. This is known as the vector sum of the two velocities.

4. Why is "plane to ground velocity" important for pilots?

Knowing the "plane to ground velocity" is important for pilots because it allows them to accurately plan flight routes and fuel consumption, as well as make adjustments for wind conditions.

5. How does "plane to ground velocity" affect flight time?

The "plane to ground velocity" can either increase or decrease the flight time, depending on the direction and strength of the wind. A headwind will slow down the aircraft and increase the flight time, while a tailwind will speed up the aircraft and decrease the flight time.

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