find the air velocity of a plane that must have a relative ground velocity of 250 km/h north if it encounters a wind pushing it toward the northeast at 75 km/h
To find the air velocity of the plane, we can use vector addition.
Let's denote:
- Vp as the velocity of the plane
- Vg as the velocity of the wind
- Vr as the relative velocity of the plane (with respect to the ground)
We are given:
- Vr = 250 km/h north (meaning the plane is flying directly north relative to the ground)
- Vg = 75 km/h towards the northeast (45 degrees east of north)
To find Vp (the air velocity of the plane), we need to find the vector sum of Vr and Vg. Let's break down the vectors into their components:
Vr = Vrx + Vry
Vg = Vgx + Vgy
Since Vr is directly north, the y-component of Vr (Vry) will be equal to 250 km/h.
Now, let's calculate the components of Vg:
We know that Vg has a magnitude of 75 km/h and is directed 45 degrees east of north. This means that:
Vgx = Vg * cos(45 degrees)
= 75 km/h * cos(45 degrees)
= 75 km/h * 0.7071
≈ 53.03 km/h (east)
Vgy = Vg * sin(45 degrees)
= 75 km/h * sin(45 degrees)
= 75 km/h * 0.7071
≈ 53.03 km/h (north)
Now, we can find the components of Vp by adding the corresponding components of Vr and Vg:
Vpx = 0 km/h + 53.03 km/h
= 53.03 km/h (east)
Vpy = 250 km/h + 53.03 km/h
= 303.03 km/h (north)
Therefore, the air velocity of the plane is approximately 53.03 km/h east and 303.03 km/h north.
To find the air velocity of the plane, we need to use vector addition. The relative ground velocity of the plane is the sum of its air velocity and the wind velocity. The air velocity and the wind velocity can be represented as vectors.
Step 1: Convert the given velocities to vectors.
The relative ground velocity of the plane is 250 km/h north, which means it has a velocity vector of 250 km/h in the north direction.
The wind is pushing the plane toward the northeast at 75 km/h, which means it has a velocity vector of 75 km/h in the northeast direction.
Step 2: Add the vectors.
To add the vectors, we need to break them down into their horizontal (x) and vertical (y) components. For the relative ground velocity vector:
- The north direction is considered the positive y-axis.
- The east direction is considered the positive x-axis.
We can represent the relative ground velocity vector as Vg = Vgx i + Vgy j, where i and j are unit vectors in the x and y directions, respectively.
Since the relative ground velocity is 250 km/h north, Vgy = 250 km/h, and Vgx = 0 km/h.
Now, let's represent the wind velocity vector as Vw = Vwx i + Vwy j.
Since the wind is pushing the plane toward the northeast, we can break it down into its horizontal and vertical components:
- The northeast direction is a combination of the north and east directions.
- The angle between the northeast and north is 45 degrees.
- The angle between the northeast and east is 45 degrees.
Using trigonometry, we can find Vwx and Vwy:
- Vwy = Vw * sin(45°)
- Vwx = Vw * cos(45°)
Step 3: Calculate the air velocity.
Now that we have the components of the relative ground velocity vector (Vgx and Vgy) and the wind velocity vector (Vwx and VWy), we can use vector addition to find the air velocity vector.
The air velocity vector (Va) can be represented as Va = Vg - Vw
Substituting the values, we get:
Va = (Vgx i + Vgy j) - (Vwx i + Vwy j)
Va = (0 km/h i + 250 km/h j) - (Vwx i + Vwy j)
Va = (0 km/h i + 250 km/h j) - (Vw * cos(45°) i + Vw * sin(45°) j)
Step 4: Calculate the air velocity magnitude and direction.
The air velocity magnitude is the magnitude of the resulting vector from the addition.
The air velocity direction is the angle between the air velocity vector and the north direction.
To get the magnitude (Va_mag) and direction (Va_dir), use the following formulas:
Va_mag = sqrt(Va_x^2 + Va_y^2)
Va_dir = atan2(Va_y, Va_x)
Using these formulas and substituting the values, you can calculate the air velocity magnitude and direction.
Keep in mind that this is a general approach to solving vector addition problems, and the specific calculations may vary based on the given values and directions.
Wind velocity vector + Air velocity vector = Ground velocity vector
Writing that as a vector equation:
75 cos 45 i + 75 sin 45 j + Vx i + Vy j = 250 j
Vx = -53.03 km/h
Vy = 250 - 53.03 = 196.97 km/h
i and j denote unit vectors in the east and north directions.
Vx and Vy are the x (east) and y (north) components of the required air velocity.
The required air speed is just 203.98 km/h