Person running in the rain with an umbrella: invariance of distances and angles under a Galilean transformation

Abstract

It is a common experience that raindrops falling vertically appear to fall at an angle to a moving person. When trying to describe the sheltered area provided by an umbrella from the viewpoints of both moving and stationary observers, at first glance it may appear that the descriptions for the sheltered area in these two frames are not identical. It is shown as well as illustrated by simulation that the sheltered areas are indeed identical in both these frames as expected. The physics behind this is explained and in the process confirming distances, angles, and consequently areas, remain invariant under a Galilean transformation. The detailed treatment of the problem given in this paper may help undergraduate students gain a robust understanding of the important concepts of frame of reference and Galilean transformation.

In the rain with and without an umbrella? The Reynolds transport theorem to the rescue

“…Addressing the question of whether it is worth running or walking in the rain (RWiR) has been the subject of a number of studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. One of the principal objectives of these studies was to determine the amount of rain that accumulates on a person during travel over a prescribed path.…”

“…Shape considerations seem to expand the range of conclusions as well. In a more recent study, Kurusingal [18] addresses for the first time the use of an umbrella while running in the rain. This study addressed primarily the viewpoints of interpreting the invariance of distances and angles under a Galilean transformation from the standpoints of a moving or stationary observers.…”