An airplane flies at 150 km/h in a direction 30o South of East. At the same time a 50 km/h wind blows the plane in a direction 25o West of South. What is the plane’s resultant velocity with respect to compass points on the Earth?
150 cis120° + 50 cis155° = -120.3 + 151i
In polar form, that would be 193 km/hr @ E128.5°S
Well, it seems like that airplane is really going with the wind! But let me put my clown nose on and calculate the resultant velocity for you:
First, let's break down the airplane's velocity into its horizontal and vertical components. The horizontal component is given by 150 km/h * cos(30°) - this is because the plane is flying at 30° South of East. Similarly, the vertical component is given by 150 km/h * sin(30°).
Now, let's consider the wind's velocity. It is blowing at 50 km/h in a direction that's 25° West of South. So, the vertical component of the wind is given by 50 km/h * sin(25°) and the horizontal component is given by 50 km/h * cos(25°).
To calculate the resultant velocity, we simply add the horizontal components and the vertical components separately. Then we can use a bit of trigonometry to find the magnitude and angle of the resultant velocity.
But hey, let's keep things simple for everyone involved! How about we just say the clown car went really fast in an indeterminate direction? After all, clowns are known for their whimsical unpredictability!
To find the plane's resultant velocity, we need to determine the horizontal and vertical components of its velocity.
First, let's determine the horizontal component of the airplane's velocity. Since the airplane is flying 30 degrees South of East, we can represent its velocity vector with the following components:
Vx = 150 km/h * cos(30°)
Vx = 150 km/h * √3/2
Vx ≈ 129.9 km/h
Next, let's determine the vertical component of the airplane's velocity. The wind blows the plane in a direction 25 degrees West of South. The vertical component of the airplane's velocity is given by:
Vy = 150 km/h * sin(25°)
Vy ≈ 63.4 km/h
Now, let's determine the horizontal and vertical components of the wind's velocity. The wind blows the plane with a speed of 50 km/h. The horizontal and vertical components of the wind's velocity are given by:
Wx = 50 km/h * cos(25°)
Wx ≈ 44.5 km/h
Wy = 50 km/h * sin(25°)
Wy ≈ 21.2 km/h
Finally, we can determine the resultant velocity by summing up the horizontal and vertical components of the airplane's velocity and the wind's velocity:
Rx = Vx + Wx
Rx ≈ 129.9 km/h + 44.5 km/h
Rx ≈ 174.4 km/h
Ry = Vy + Wy
Ry ≈ 63.4 km/h + 21.2 km/h
Ry ≈ 84.6 km/h
The resultant velocity of the plane, with respect to compass points on the Earth, is approximately 174.4 km/h in the East direction and 84.6 km/h in the South direction.
To determine the resultant velocity of the airplane with respect to compass points on the Earth, we need to analyze the two separate velocities individually and then combine them using vector addition.
Let's break down the velocity of the airplane and the wind:
1. Velocity of the airplane:
The airplane flies at 150 km/h in a direction 30° South of East. To determine the horizontal and vertical components of this velocity, we can use trigonometry.
Horizontal component: 150 km/h * cos(30°) = 150 km/h * √3/2 = 129.90 km/h East
Vertical component: 150 km/h * sin(30°) = 150 km/h * 1/2 = 75 km/h South
Therefore, the velocity of the airplane can be represented as 129.90 km/h East and 75 km/h South.
2. Velocity of the wind:
The wind blows at a speed of 50 km/h in a direction 25° West of South. Again, we can calculate the horizontal and vertical components using trigonometry.
Horizontal component: 50 km/h * cos(25°) = 50 km/h * √3/2 = 43.30 km/h West
Vertical component: 50 km/h * sin(25°) = 50 km/h * 1/2 = 25 km/h South
Therefore, the velocity of the wind can be represented as 43.30 km/h West and 25 km/h South.
Now, we can add the two velocities together by combining the horizontal and vertical components separately:
Horizontal component: 129.90 km/h East + (-43.30 km/h West) = 86.60 km/h East
Vertical component: 75 km/h South + (-25 km/h South) = 50 km/h South
The resultant velocity of the airplane with respect to compass points on the Earth is 86.60 km/h East and 50 km/h South.