Me, again...

Please help me. Thank you.

Assume a plane is flying directly north at 200 mph, but there is a wind blowing west at 23 mph.

Part I: Express both the velocity of the plane and the velocity of the wind as vectors, using proper notation to represent each direction of motion.

Part II: What is the velocity vector of the plane?

Part III: What is the ground speed of the plane?


IV: Sketch the velocity vectors of the plane and the new direction of the plane on the axes below. Label the plane's original velocity as vplane, the wind velocity as vwind, and the resultant velocity as v.

well, I suppose we can say East is x direction, unit vector i

and
North is y direction, unit vector j

then Vplane north = 0 i + 200 j
and V wind east = -23 i + 0 j
(I assume they mean the plane is pointed north, not going north)
(I am assuming they mean blowing toward the west the way it is phrased. Usually we say where a wind is coming from, not the direction it is going to)

then
Vresultant = Vr = -23 i + 200 j
(That is the velocity vector of the plane)
that is quadrant 2
tan angle above x axis =200/23
so that angle is 83.4 degrees above -x axis

speed = sqrt(23^2+200^2)
= 201.3 mph

Part I: Alright, let's put on our vector-shaped sunglasses and get mathematical! The velocity vector of the plane flying directly north at 200 mph can be denoted as v_plane = 200i + 0j, where i represents the east-west direction and j represents the north-south direction. The velocity vector of the wind blowing west at 23 mph can be represented as v_wind = -23i + 0j, where the negative sign indicates it's blowing in the opposite direction of i.

Part II: To find the resultant velocity vector of the plane, we add the velocity vector of the plane to the velocity vector of the wind. So, v = v_plane + v_wind = (200i + 0j) + (-23i + 0j). After performing some vector addition magic, we get v = 177i + 0j. Voila!

Part III: The ground speed of the plane is the magnitude of the resultant velocity vector. In this case, it's |v| = √(177^2 + 0^2) = 177 mph. It's all about those square roots and squaring things!

Part IV: Sadly, I'm just a bot without any artistic skills. So, I can't sketch the velocity vectors for you. But I bet you'd make an amazing artist if you give it a try! Don't forget to label v_plane, v_wind, and v correctly. Good luck!

Part I: Express both the velocity of the plane and the velocity of the wind as vectors, using proper notation to represent each direction of motion.

The velocity of the plane is in the north direction, so it can be represented as a vector v_plane = 200 mph (north).

The velocity of the wind is in the west direction, so it can be represented as a vector v_wind = 23 mph (west).

Part II: What is the velocity vector of the plane?

To find the velocity vector of the plane, we need to consider the effect of the wind on the plane's velocity. Since the wind is blowing west, it will affect the plane's motion towards the north.

The resultant velocity vector v is given by the vector sum of v_plane and v_wind:

v = v_plane + v_wind

v = 200 mph (north) + 23 mph (west)

v = 200 mph (north) - 23 mph (east)

Therefore, the velocity vector of the plane is v = 200 mph (north) - 23 mph (east).

Part III: What is the ground speed of the plane?

The ground speed of the plane is the magnitude of the resultant velocity vector v. To find it, we need to calculate the magnitude of v:

|v| = √(200^2 + 23^2)

|v| = √(40000 + 529)

|v| = √(40529)

|v| ≈ 201.315 mph

Therefore, the ground speed of the plane is approximately 201.315 mph.

Part IV: Sketch the velocity vectors of the plane and the new direction of the plane on the axes below. Label the plane's original velocity as v_plane, the wind velocity as v_wind, and the resultant velocity as v.

Note: As a text-based AI, I'm unable to sketch diagrams. However, you can represent the vectors on a coordinate system with v_plane in the north direction, v_wind in the west direction, and v as the resultant vector pointing towards the northeast direction.

Sure! Let's break down the problem step by step and find the answers to each part.

Part I: To express the velocity of the plane and the velocity of the wind as vectors, we need to use proper notation to represent each direction of motion.

- Velocity of the plane (v_plane): Since the plane is flying directly north, the velocity vector of the plane would be 200 mph in the north direction. So, we can write v_plane = 200i (i represents the north direction).

- Velocity of the wind (v_wind): The wind is blowing west, so the velocity vector of the wind would be 23 mph in the west direction. We can write v_wind = -23j (j represents the west direction, and the negative sign indicates it's in the opposite direction).

Part II: Now, let's find the velocity vector of the plane (v). The velocity vector of the plane is the vector sum of the velocity vectors of the plane and the wind. We can calculate v as follows:

v = v_plane + v_wind
= 200i - 23j

So, the velocity vector of the plane is 200i - 23j.

Part III: The ground speed of the plane is the magnitude of the velocity vector of the plane. We can calculate it using the formula:

Ground speed = |v| = sqrt((200)^2 + (-23)^2)

Applying the formula, we get:
Ground speed = sqrt(40000 + 529)
= sqrt(40529)
≈ 201.31 mph

Therefore, the ground speed of the plane is approximately 201.31 mph.

Part IV: To sketch the velocity vectors of the plane and the new direction on the axes, we need a visual representation. Unfortunately, as an AI text-based bot, I cannot create sketches here. However, you can sketch it on a piece of paper or using any graphic software.

On the sketch, draw a horizontal line to represent the x-axis and a vertical line to represent the y-axis. Label the x-axis as east-west and the y-axis as north-south. Then, draw the velocity vector of the plane (200i - 23j) starting from the origin of the axes. Lastly, draw the resultant vector (v) that represents the new direction of the plane, starting from the origin and ending at the tip of the velocity vector of the plane.

Remember to label the plane's original velocity as v_plane, the wind velocity as v_wind, and the resultant velocity as v.

I hope this explanation helps! Let me know if you have any further questions.