Question

A room has dimensions 3.00 m (height) 3.70 m 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.)

Answer 1

The displacement is the straight-line distance it flew when everything is over.  We don't count all of the turns that it made in the displacement.  So if it's in an opposite corner we just need to know how far it is from one corner of a box to the opposite corner with the given dimensions.  We use the Pythagorean Theorem for part a):
$d= \sqrt{3.00^{2}+3.70^{2}+4.30^{2}}=6.42m$
b).  Since the displacement is the straight line path, there are no shorter paths that this.
c)All other paths are greater
d) if the fly flies this path, then they would be equal
in cartesian coordinates, the vector is the sum of the 3 components whose magnitudes are the wall lengths and whose directions are parallel to that wall:
$D=a_x3.70+a_y4.30+a_z3.00$

The shortest walking path is 7.96m.