A plane is steering at S83degreesE at an air speed of 550 km/h. The wind is from N70degreesE at 85km/h. Find the ground velocity of the plane to 1 decimal space.
the vectors to add are
(550cos7°,550sin7°)+(85cos20°,85sin20°)
To find the ground velocity of the plane, we need to consider the effect of the wind on the plane's motion.
First, we need to decompose the velocity vectors of the plane and the wind into their North and East components.
The velocity of the plane:
- Direction: S83°E
- Speed: 550 km/h
To find the North and East components, we can use trigonometry. Let's denote the North and East components as Np and Ep, respectively.
Np = speed * sin(direction)
Np = 550 * sin(83°) ≈ 541.518 km/h (North component of the plane's velocity)
Ep = speed * cos(direction)
Ep = 550 * cos(83°) ≈ 136.743 km/h (East component of the plane's velocity)
The velocity of the wind:
- Direction: N70°E
- Speed: 85 km/h
Again, let's find the North and East components of the wind velocity. Denote them as Nw and Ew, respectively.
Nw = speed * sin(direction)
Nw = 85 * sin(70°) ≈ 79.862 km/h (North component of the wind's velocity)
Ew = speed * cos(direction)
Ew = 85 * cos(70°) ≈ 30.424 km/h (East component of the wind's velocity)
Now that we have the North and East components for both the plane and the wind, we can find the resultant velocity (ground velocity) vector by adding the corresponding components together.
Ng = Np + Nw ≈ 541.518 + 79.862 ≈ 621.38 km/h (North component of the ground velocity)
Eg = Ep + Ew ≈ 136.743 + 30.424 ≈ 167.167 km/h (East component of the ground velocity)
Finally, use the Pythagorean theorem to calculate the magnitude of the ground velocity (resultant velocity):
Ground velocity = √(Ng² + Eg²)
Ground velocity ≈ √(621.38² + 167.167²) ≈ √(386014.664 + 27974.389) ≈ √(414989.053) ≈ 643.5 km/h
Therefore, the ground velocity of the plane is approximately 643.5 km/h (rounded to 1 decimal place).