An airplane has an airspeed of 400 km and is heading due west. If it encounters wind blowing north at 120 km/h, how many degrees north from west is the resultant velocity of the plane?
tan(Θ) = 120 / 400
To solve this problem, we can use vector addition. We need to find the resultant velocity of the airplane by considering the airspeed and the wind velocity.
1. Let's start by drawing a diagram. We'll draw a horizontal line to represent the west direction and a vertical line to represent the north direction.
N
+-------+
| |
| |
W | |
| |
+-------+
S
2. We know that the airplane has an airspeed of 400 km/h, which is directed west. We represent this velocity as a vector with a length of 400 km/h, pointing to the left (west direction).
N
+-------+
| |
| |
400 | <-----
km/h | |
+-------+
S
3. The wind is blowing north at 120 km/h. We represent this wind velocity as a vector with a length of 120 km/h, pointing upwards (north direction).
<---- N
+-------+
---->| |
120 | |
km/h | |
+-------+
S
4. To find the resultant velocity, we add these two vectors together. We can use the Pythagorean theorem to find the magnitude of the resultant velocity.
magnitude of resultant velocity = sqrt((400 km/h)^2 + (120 km/h)^2) = sqrt(160000 + 14400) = sqrt(174400) ≈ 417.65 km/h
5. Now, to find the direction of the resultant velocity, we can use trigonometry. The tangential component of the velocity is given by the westward component, which is 400 km/h. The normal component of the velocity is given by the northward component, which is 120 km/h.
tangent of the angle = northward component / westward component = 120 km/h / 400 km/h ≈ 0.3
angle = atan(0.3) ≈ 16.7 degrees
6. Therefore, the resultant velocity of the plane is approximately 417.65 km/h, and the angle of the resultant velocity from the west direction is approximately 16.7 degrees north.
To find the direction of the resultant velocity of the plane, we need to use vector addition.
First, let's represent the airplane's westward velocity as a negative vector (-400 km/h) and the wind's northward velocity as a positive vector (+120 km/h).
Next, we can add these two vectors geometrically by creating a parallelogram. Draw a horizontal line representing the airplane's westward velocity (-400 km/h), and then draw a vertical line representing the wind's northward velocity (+120 km/h) starting from the end of the horizontal line.
Now, draw a diagonal line connecting the starting point (tail of westward velocity vector) to the ending point (tip of the northward velocity vector). This diagonal line represents the resultant velocity of the airplane.
To calculate the magnitude of the resultant velocity, we can use the Pythagorean theorem. The magnitude of the westward velocity is 400 km/h, and the magnitude of the northward velocity is 120 km/h. So, the magnitude of the resultant velocity (V) can be found using the equation:
V^2 = (-400 km/h)^2 + (120 km/h)^2
V^2 = 160,000 km^2/h^2 + 14,400 km^2/h^2
V^2 = 174,400 km^2/h^2
V ≈ 417.818 km/h
To find the direction of the resultant velocity, we can use trigonometry. The angle (θ) between the resultant velocity vector and the westward direction can be found using the following equation:
tan(θ) = opposite/adjacent
tan(θ) = 120 km/h / 400 km/h
θ = atan(120/400)
θ ≈ 16.699 degrees
So, the resultant velocity of the plane is approximately 417.818 km/h at an angle of approximately 16.699 degrees north from west.