A plane flies on a heading of S60 degrees East at a constant speed of 550 km/h.
If the velocity of the wind is 50 km/h on a bearing of S40 degrees West, what is the velocity of the plane with respect to the ground?
That is if you know if the terminology means the wind is coming from the southwest or heading to the southwest. Bearing does not mean heading or direction of travel. I have no idea which it means. For me a SW wind means from the SW headed NE
v = 550(cos330,sin330) + 50(cos230,sin230)
= (476.3139..., -275) + (-32.1393..., -38.30222...)
= (444.17459..., -313.022..)
|v| = √(444.17459...^2 + (-313.022..)^2) = appr 543.55
or by cosine Law after making your sketch and using the diagonal of the parallelogram.
v^2 = 550^2 + 50^2 - 2(550)(50)cos80°
= 543.55..
the resultant velocity is just the sum of the two vectors. To get that, convert each vector to x- and y- coordinates, add them, then convert back to polar form.
To find the velocity of the plane with respect to the ground, we need to consider the effect of the wind on the plane's motion. We can break down the given information as follows:
1. The plane is flying on a heading of S60 degrees East at a constant speed of 550 km/h. This means the plane is moving in the direction of S60 degrees East at a speed of 550 km/h.
2. The wind is blowing with a velocity of 50 km/h on a bearing of S40 degrees West. This means the wind is blowing in the direction of S40 degrees West at a speed of 50 km/h.
To determine the velocity of the plane with respect to the ground, we can use vector addition. We can break down the plane's velocity and the wind's velocity into their respective components:
Plane's velocity:
- In the x-direction: 550 km/h * cos(60°) = 275 km/h
- In the y-direction: 550 km/h * sin(60°) = 475 km/h
Wind's velocity:
- In the x-direction: 50 km/h * cos(40°) = 38.216 km/h
- In the y-direction: 50 km/h * sin(40°) = 32.086 km/h
Now, we can add the respective components to find the resultant velocity of the plane with respect to the ground:
Resultant velocity in the x-direction: 275 km/h + 38.216 km/h = 313.216 km/h
Resultant velocity in the y-direction: 475 km/h + 32.086 km/h = 507.086 km/h
Using the Pythagorean theorem, we can calculate the magnitude of the resultant velocity:
Resultant velocity = sqrt((313.216 km/h)^2 + (507.086 km/h)^2) = 603.838 km/h
The direction of the resultant velocity can be found using trigonometry:
Resultant velocity angle = arctan(507.086 km/h / 313.216 km/h) = 59.731 degrees
Therefore, the velocity of the plane with respect to the ground is approximately 603.838 km/h at an angle of 59.731 degrees.