1) Imagine you were to walk 12 km 25 degrees North of East. If you could only walk south and west to get back to the point that you started from, how far would you have to walk?
2) An airplane has a velocity of 70 m/s [50 degrees W of N] with respect to the air. A person on the ground looks up and observes velocity of the airplane to be 60 m/s [20 degrees N of W], what was the velocity of the wind?
Help with either of these question would be greatly appreciated.
walk W for 12cosine(25), then S for 12*Sin(25)
Sure! Let's solve these questions step by step.
1) To find the distance you have to walk to get back to your starting point, we first need to break down the initial walk into its components.
You walked 12 km at 25 degrees North of East. This can be broken down into two components:
a) The North component: To find this, we need to determine the distance you walked directly towards the North. We can use trigonometry to calculate this. Since you walked at 25 degrees North of East, the angle between your North component and the Eastward direction is 90 - 25 = 65 degrees. Using trigonometric ratios, you can find the North component as follows:
North component = 12 km * sin(65°)
b) The East component: To find this, we need to determine the distance you walked directly towards the East. Again, we can use trigonometry to calculate this. Since you walked at 25 degrees North of East, the angle between your East component and the Eastward direction is 25 degrees. Using trigonometric ratios, you can find the East component as follows:
East component = 12 km * cos(25°)
Now, to find the total distance you have to walk to get back to your starting point, we need to consider the opposite directions of the North and East components.
The distance you have to walk South is equal to the North component, and the distance you have to walk West is equal to the East component.
Therefore, the total distance you have to walk is given by:
Total distance = North component + West component
= North component + East component
Note that we don't include the East component in the equation since it is in the opposite direction, which is West.
2) To find the velocity of the wind, we can apply the concept of vector addition. The velocity of the airplane with respect to the ground is the vector sum of the velocity of the airplane with respect to the air and the velocity of the wind.
Let's assume the velocity of the wind is represented as W m/s [θ degrees W of N].
The velocity of the airplane with respect to the ground is 60 m/s [20 degrees N of W]. This velocity can be broken down into two components:
a) The North component: To find this, we can use trigonometry. Since the angle is 20 degrees N of W, the angle between the North component and the Westward direction is 20 degrees. Using trigonometric ratios, we can find the North component as follows:
North component = 60 m/s * sin(20°)
b) The West component: To find this, we can also use trigonometry. Since the angle is 20 degrees N of W, the angle between the West component and the Westward direction is 90 - 20 = 70 degrees. Using trigonometric ratios, we can find the West component as follows:
West component = 60 m/s * cos(20°)
Now, we can write the vector equation for the velocity of the airplane with respect to the ground:
Velocity of airplane with respect to ground = Velocity of airplane with respect to air + Velocity of wind
By comparing the corresponding components, we can set up the following equations:
North component of airplane velocity = North component of airplane-air velocity + North component of wind velocity
West component of airplane velocity = West component of airplane-air velocity + West component of wind velocity
Substituting the given values and solving these equations simultaneously will allow us to find the velocity of the wind (W) and its direction (θ).