while flying due west at 1.20 x 10^2 km/h, an airplane is blown due north at 45.0 km/h by the wind. What is the magnitude and direction (relative to 000) of the plane's resultant velocity in km/h?
To find the magnitude and direction of the plane's resultant velocity, we can use vector addition.
Let's break down the given information into vectors:
1. The plane's velocity due west: 1.20 x 10^2 km/h in the west direction.
2. The wind's velocity due north: 45.0 km/h in the north direction.
To find the resultant velocity, we need to add these two vectors together.
Step 1: Convert the velocities into vector form.
1. Plane's velocity due west:
- Magnitude: 1.20 x 10^2 km/h
- Direction: 180° (opposite direction to the east)
2. Wind's velocity due north:
- Magnitude: 45.0 km/h
- Direction: 90° (north)
Step 2: Add the vectors using vector addition.
To add the vectors, we need to break them down into their components.
- Plane's velocity due west: -1.20 x 10^2 km/h in the x-direction (west).
- Wind's velocity due north: +45.0 km/h in the y-direction (north).
Next, add the components of the vector:
X-component = -1.20 x 10^2 km/h (west)
Y-component = +45.0 km/h (north)
Step 3: Calculate the resultant velocity.
To find the resultant velocity, we can use the Pythagorean theorem and trigonometry.
Resultant magnitude = √[(X-component)^2 + (Y-component)^2]
Plugging in the values:
Resultant magnitude = √[(-1.20 x 10^2)^2 + (45.0)^2] km/h
= √[14400 + 2025] km/h
= √16425 km/h
≈ 128.17 km/h
Step 4: Calculate the direction of the resultant velocity.
To find the direction, we can use the inverse tangent (arctan) function.
Resultant direction = arctan(Y-component / X-component)
Plugging in the values:
Resultant direction = arctan(45.0 km/h / -1.20 x 10^2 km/h)
= arctan(-0.375)
Using a calculator, we find that the arctan(-0.375) is approximately -20.56°.
Since the plane is flying in the west direction, the direction of the resultant velocity relative to 000 is 180° - 20.56° = 159.44°.
Therefore, the magnitude of the plane's resultant velocity is approximately 128.17 km/h, and its direction relative to 000 is approximately 159.44°.