Galileo's kinematical paradox and the expanding circle of simultaneity

Abstract

No abstract

Galileoʼs kinematical paradox and the role of resistive forces

“…A similar argument shows that the descent time along chord BC is the same for any choice of B, which was the original formulation of the paradox by Galileo in his letter to Guidobaldo del Monte [1]. Another way of presenting Galileoʼs kinematical paradox is, as we have already mentioned, through the expanding circle of simultaneity [5]. Consider the Cartesian axes shown in figure 1.…”

“…Galileoʼs kinematical paradox has an interesting consequence: if several bodies initially at rest are released simultaneously from the top of the circle, each one sliding along a different chord starting at A (like AB in figure 1), then at a given instant their positions will be on an expanding circle whose centre moves down with acceleration g 2. This was already discussed by Galileo in [2], and a more pedagogically oriented presentation can be found in [5,6].…”

“…It is easily shown (see [5]) that equations (5) and (6) which represents a circle with time-dependent radius given by…”

“…Figure 2 shows the simultaneity circle for different values of τ t . Links to video demonstrations of these effects can be found in [5,6].…”

“…It is easily shown (see [5]) that equations ( 5) and ( 6) lead to The centre of this circle moves vertically along the diameter AC with acceleration g 2. This means that, at a given time, particles starting from A along different chords all lie on a circle, the synchronous curve for this problem.…”