How Do You Calculate the Fly's Displacement in a 3D Room?

[vysero]

I was not sure what the title of this thread should be. This should probably give you an idea of how little I understand this problem:

A room has dimensions 3.00 m (height) x 3.70 m x 4.30 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. A) What is the magnitude of its displacement? B) Could the length of its path be less than this magnitude? C) Greater? D) Equal? E) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. F) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

First big problem, does it go height, width, length or is it the other way around?

 

[CAF123]

I was not sure what the title of this thread should be. This should probably give you an idea of how little I understand this problem:

A room has dimensions 3.00 m (height) x 3.70 m x 4.30 m. A fly starting at one corner flies around, ending up at the diagonally opposite corner. A) What is the magnitude of its displacement? B) Could the length of its path be less than this magnitude? C) Greater? D) Equal? E) Choose a suitable coordinate system and express the components of the displacement vector in that system in unit-vector notation. F) If the fly walks, what is the length of the shortest path? (Hint: This can be answered without calculus. The room is like a box. Unfold its walls to flatten them into a plane.)

First big problem, does it go height, width, length or is it the other way around?

What have you tried? The displacement vector will be from the fly's initial position to it's final position. The magnitude of this will be $|\vec{d}| = \sqrt{d_x^2 + d_y^2 + d_z^2}$. Some suggestions: let the x, y and z axis represent the length, depth and height of the room respectively. For the later part of the problem, know the triangle inequality; that is $|x| + |y| ≥ |x + y|$, ie the sum of two sides of a triangle is always greater than the hypotenuse.

 

[vysero]

Ok so the displacement vector is equal to the square of the sum of LxWxH^1/2. That comes too 6.42 which is the correct answer. Now my original problem they label the height as 3 m how do I know what the other to values are? Like for instance is 3.7 m the length or the width of the room?

For B) I am assuming the answer is No and therefore I would assume that C) would also be No and that D) would be Yes. For the last two questions I am am a bit confused on where to start.

 

[voko]

Why can't the path be greater?

 

[Nickie]

I can provide a response to this content by breaking down the problem and providing some guidance on how to approach it.

Firstly, the dimensions of the room are given as 3.00 m (height) x 3.70 m x 4.30 m. This means that the height of the room is 3.00 m, the width is 3.70 m, and the length is 4.30 m. It is important to note the order of the dimensions as it will be relevant in solving the problem.

Next, we are told that a fly starts at one corner and flies around, ending up at the diagonally opposite corner. This means that the fly's path forms a diagonal line across the room.

Now, let's address the questions that are asked:

A) What is the magnitude of its displacement?

To answer this question, we need to understand what displacement means. Displacement is the shortest distance between the starting point and the ending point. In this case, the starting and ending points are the two corners of the room. So, the magnitude of the fly's displacement is the length of the diagonal line that connects the two corners. To find this, we can use the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (the diagonal line) is equal to the sum of the squares of the other two sides. In this case, the other two sides are the height and the width of the room. So, the magnitude of the fly's displacement can be calculated as:

√(3.00 m)^2 + (3.70 m)^2 + (4.30 m)^2 = 6.93 m

B) Could the length of its path be less than this magnitude?

No, the length of the path cannot be less than the magnitude of the displacement. As mentioned earlier, displacement is the shortest distance between two points, so the length of the path must be equal to or greater than the magnitude of the displacement.

C) Greater?

Yes, the length of the path could be greater than the magnitude of the displacement. This would happen if the fly does not fly in a straight line and instead takes a longer path around the room.

D) Equal?

Yes, the length of the path could also be equal to the magnitude of the displacement. This would happen if the fly flies in a straight line from one corner to