a plane with an airspeed of 192mi/h is headed on a bearing of 121 degrees. a north wind is blowing (from north to south) at 15.9 mi/h . find the ground speed and the actual bearing of plane.
I will assume you are using the standard 'true north' reference and a clockwise rotation resulting a vector in the fourth quadrant with a magnitude of 192.
draw a vector from the end of the previous vector due south with a magnitude of 15.9
draw in the resultant vector
by cosine law:
R^2 = 192^2 + 15.9^2 - 2(192)(15.9)cos 121º
R will be the ground speed.
Use the Sine Law to find the angle across from the 15.9 side, add that angle to 121 and you have your actual bearing.
is it gonna be right traingle or what?
*
* *
15.9 * 121 *
* *
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**************
192
10.76
Well, this plane seems to have quite the comedy of errors going on. Let's see if we can sort it out.
To find the ground speed, we need to take into account both the airspeed and the wind speed. It's like trying to run against a strong wind – it slows you down.
Using a bit of trigonometry, we can say that the ground speed is the vector sum of the airspeed and wind speed.
So, to find the ground speed, we square the airspeed and the wind speed, add them together, and then take the square root of the sum.
Let's do some math. 192 squared is 36,864. 15.9 squared is 252.81. Adding those together gives us 37,116.81. And taking the square root of that gives us approximately 192.46.
So, the ground speed of the plane is approximately 192.46 mi/h.
Now, to find the actual bearing, we need to take into account both the initial bearing and the effect of the wind.
To do that, we use a little bit more trigonometry. We can use the wind speed and the ground speed to find the angle between the ground track and the original bearing.
Let's calculate. The wind speed is 15.9 mi/h and the ground speed is 192.46 mi/h. So, the ratio of the wind speed to the ground speed is approximately 0.0826.
Using an inverse tangent function, we can find the angle whose tangent is 0.0826. That gives us approximately 4.72 degrees.
Now, to find the actual bearing, we simply subtract that angle from the original bearing.
So, the actual bearing of the plane is approximately 121 - 4.72 = 116.28 degrees.
There you have it, the ground speed is approximately 192.46 mi/h and the actual bearing is approximately 116.28 degrees. Bon voyage, little plane!
To find the ground speed and actual bearing of the plane, we need to break down the vector components of the airspeed and wind velocity. Let's go step by step:
1. Convert the airspeed of 192 mi/h into its vector components. The airspeed will form a right-angled triangle with the east-west component being the adjacent side and the north-south component being the opposite side. We can use trigonometry to calculate these components:
East-West component = airspeed * cos(bearing)
= 192 mi/h * cos(121°)
North-South component = airspeed * sin(bearing)
= 192 mi/h * sin(121°)
Calculate these values using a scientific calculator.
2. Now, let's determine the ground speed of the plane. The ground speed is the vector sum of the airspeed and wind velocity. Since the wind is blowing from north to south, it will affect the north-south component of the plane's motion. The east-west component will remain unchanged. Let's calculate the new vector components:
Ground East-West component = East-West component of airspeed
= calculated in step 1
Ground North-South component = North-South component of airspeed - wind velocity
= calculated in step 1 - 15.9 mi/h
Calculate these values.
3. To find the ground speed, we need to calculate the magnitude of the ground velocity vector using the Pythagorean theorem:
Ground Speed = √(Ground East-West component^2 + Ground North-South component^2)
4. Lastly, we need to find the bearing of the plane with respect to the ground. We can use trigonometry once again:
Actual Bearing = arctan(Ground East-West component / Ground North-South component)
Calculate this using the values obtained in step 2.
By following these steps, you should be able to find the ground speed and actual bearing of the plane.