The problem is: an airplane is flying in the direction 120 degrees with an airspeed of 320 mph, and a 35 mph wind is blowing in the direction 60 degrees. The question ask to find the true course and the ground speed. To solve the problem I have done this:
Flying 120sin 320_+120cos320_
Wind 60sin(35-180)_+60cos(35-180)_
I understand that I then add the together. The problem is where there are _ I don't know what direction I am suppost to put and how to determine it.
(320cos120 , 320sin120) + (35cos60 , 35sin60)
you should be familiar with the trig ratios of the 30-60-90 triangle, and the multiples of those angles using the CAST rule
= (320(-1/2) , 320(√3/2) + (35(1/2) + 35(√3/2)
= (-160 , 160√3) + (35/2 , 35√3/2)
=( -142.5 , 177.5√3)
magnitude = √( (-142.5)^2 + (177.5√3)^2 )
= appr 338.9 mph
direction:
tanØ = 177.5√3/-142.5 = -2.157...
Ø = 180 - 65.13 = appr 114.9°
I you make a sketch you will see that you could also use the cosine law,
I got the same answer that way.
To solve this problem, you need to break down the given information and apply some trigonometry concepts.
1. Determine the components of the airplane's velocity:
- The airspeed of the airplane is given as 320 mph, with a direction of 120 degrees. To find the horizontal and vertical components of this velocity, you can use trigonometry. The horizontal component is calculated as airspeed * cosine(direction), and the vertical component is calculated as airspeed * sine(direction). In this case, the horizontal component would be 320 * cos(120°), and the vertical component would be 320 * sin(120°). Make sure to use the appropriate units for angle measurement (degrees or radians).
- If you perform these calculations, you will find that the horizontal component of the airplane's velocity is -160 mph, and the vertical component is -276.8 mph (rounded to one decimal place). The negative sign indicates the direction to the west (considered negative in this example).
2. Determine the components of the wind velocity:
- The wind speed is given in the problem as 35 mph, with a direction of 60 degrees. Similarly, you can use trigonometry to find the horizontal and vertical components of the wind velocity. The horizontal component can be found as wind speed * cosine(direction) and the vertical component as wind speed * sine(direction). In this case, the horizontal component would be 35 * cos(60°), and the vertical component would be 35 * sin(60°). Again, ensure you use the appropriate units for angle measurement.
- If you perform these calculations, you will find that the horizontal component of the wind velocity is 17.5 mph, and the vertical component is 30.2 mph (rounded to one decimal place).
3. Find the total components of velocity:
- To find the actual velocity, you need to add the corresponding components of the airplane's velocity and the wind velocity in both horizontal and vertical directions. Add the horizontal components to find the total horizontal velocity, and add the vertical components to find the total vertical velocity.
- In this case, the total horizontal velocity can be calculated as the sum of the airplane's horizontal velocity (-160 mph) and the wind's horizontal velocity (17.5 mph). Similarly, the total vertical velocity can be calculated as the sum of the airplane's vertical velocity (-276.8 mph) and the wind's vertical velocity (30.2 mph).
4. Calculate the true course and ground speed:
- Now, you can calculate the true course (direction of the resultant velocity) and the ground speed (magnitude of the resultant velocity).
- The true course can be found using the formula: angle = arctan(vertical velocity / horizontal velocity). In this case, the true course would be arctan(total vertical velocity / total horizontal velocity).
- The ground speed can be calculated as the magnitude of the resultant velocity, which can be determined using the Pythagorean theorem: magnitude = sqrt(horizontal velocity^2 + vertical velocity^2). In this case, the ground speed would be sqrt(total horizontal velocity^2 + total vertical velocity^2).
By following these steps, you should be able to find the true course and ground speed of the airplane.