A plane is flying south east at a constant speed of 900km/h. The wind is blowing towards the north at 100km/h. Determine the resultant velocity of the plane relative to the ground.
the resultant is
<900/√2,-900/√2 + 100>
Now just find the speed and direction.
To determine the resultant velocity of the plane relative to the ground, you need to calculate the vector sum of the plane's velocity and the wind's velocity.
First, let's break down the given information into vectors. The plane's velocity can be represented as a vector pointing southeast with a magnitude of 900 km/h. The wind's velocity can be represented by a vector pointing north with a magnitude of 100 km/h.
To find the resultant velocity, we need to add these two vectors together. Since the vectors are not in the same direction, we'll use vector addition.
We can create a right-angled triangle where the plane's velocity vector is the hypotenuse and the wind's velocity vector is one of the legs. By applying the Pythagorean theorem, we can find the magnitude of the resultant velocity.
Let's denote the magnitude of the resultant velocity as V and the angle between the wind's velocity vector and the plane's velocity vector as θ.
Using the Pythagorean theorem, we have:
V^2 = (900 km/h)^2 + (100 km/h)^2
V^2 = 810,000 + 10,000
V^2 = 820,000
V = √820,000 km/h
V ≈ 906.54 km/h
So, the magnitude of the resultant velocity, relative to the ground, is approximately 906.54 km/h.
To determine the direction of the resultant velocity vector, we'll use trigonometry. Since the wind is blowing towards the north, and the plane is flying southeast, the angle θ can be expressed as the complement of the angle between the northeast direction and the wind's direction.
Let's denote the direction of the resultant velocity vector as φ.
tan(θ) = (900 km/h) / (100 km/h)
θ ≈ arctan(900/100)
θ ≈ 84.29°
To get φ, we subtract θ from 45° (since southeast is 45° clockwise from north):
φ = 45° - θ
φ ≈ 45° - 84.29°
φ ≈ -39.29°
Therefore, the resultant velocity of the plane relative to the ground is approximately 906.54 km/h in a direction of 39.29° west of south.