Tom flies the helicopter through calm skies at a
velocity v1. The direction of v1 is 15o north of east,
and the speed is 180 km/h.
Eventually, however, he enters a windy part of the
atmosphere and finds that his helicopter now
moves at a velocity v2. The direction of v2 is due
east, and the speed is 150 km/h.
Assuming that ‘due east’ is 0o, ‘due north’ is 90o,
and so on, find
i. the speed of the wind;
ii. in what direction is the wind blowing
If the wind speed is <x,y> then you have
<180 sin15°,180 cos15°> + <x,y> = <150,0>
Now you can find x and y, and thus the speed and direction.
To find the speed of the wind and in what direction it is blowing, we need to analyze the velocity vectors and use trigonometric relationships.
Let's break down the problem step by step:
Step 1: Draw a diagram
Draw a diagram representing the velocity vectors v1 and v2. Use an appropriate scale to represent the magnitudes of the velocities.
Step 2: Analyze the known information about v1
The direction of v1 is 15° north of east, and its speed is 180 km/h.
Step 3: Analyze the known information about v2
The direction of v2 is due east, and its speed is 150 km/h.
Step 4: Decompose the velocity vectors
Decompose the velocity vectors v1 and v2 into their horizontal (x) and vertical (y) components.
For v1:
The x-component (v1x) can be found using the cosine of the angle between v1 and the positive x-axis, which is (90° - 15°) since v1 is 15° north of east.
v1x = v1 * cos(90° - 15°)
The y-component (v1y) can be found using the sine of the angle between v1 and the positive x-axis.
v1y = v1 * sin(90° - 15°)
For v2:
Since v2 is due east, its x-component (v2x) is the total velocity, which is 150 km/h.
v2x = 150 km/h
The y-component (v2y) of v2 would be zero since it does not have any vertical component.
Step 5: Determine the wind velocity components
The wind is represented by the difference between v2 and v1, assuming the wind is affecting the helicopter's motion only in the horizontal direction (i.e., x-component).
The x-component of the wind velocity is the difference between the x-components of v2 and v1.
wind velocity in x-direction = v2x - v1x
To solve for each component, plug in the values for v1x and v2x.
The y-component of the wind velocity would be zero since the problem does not mention any alteration to the vertical component.
Step 6: Calculate the magnitude and direction of the wind velocity
Using the components of the wind velocity, we can calculate the magnitude and direction of the wind velocity.
The magnitude of the wind velocity can be found using Pythagorean's theorem:
magnitude of the wind velocity = sqrt((wind velocity in x-direction)^2 + (wind velocity in y-direction)^2)
To find the direction of the wind, use the inverse tangent function:
direction of the wind = arctan(wind velocity in y-direction / wind velocity in x-direction)
Step 7: Substitute the values obtained in Step 6 back into the given units (km/h) to complete the answer.
Following these steps will allow you to find the speed of the wind and the direction in which it is blowing based on the information provided.