Living on a line

One or two months ago I came to an interesting story which shows how geometry can often transform your understanding of different physical effects.

Circles

This story starts like this :

” Imagine you are a tiny physicist living on a strait line. What you do not know, is that the line on which you live, let’s call it line-land, is part of a bigger space: a 2D flat surface. Let’s face it, the universe on which you stand is boring. There is not much happening on line-land. The only thing living on the plane are circles.  These circles are traveling with a constant velocity and from time to time they enter in contact with your world (the line on which you live). When a circle crosses the line, the tiny physicist that you are can observe the intersection points (between the line and the circle)  moving as the circle is passing by. What you observe is the sudden creation of two points moving in opposite direction, until the distance between them is exactly equal to the circle diameter. Then, the points are moving towards each other. They finally enter in collision and annihilate. These points, you call them particles. In your world, these could even be your elementary particles. However, for you the story is not that simple, simply because you are not aware of this extra dimension in which your line stands. Let me remind you that you believe that the entire universe is what you can see. To you : “the universe is the line”. What you know however, is Newton’s law : F=ma. And from this equation, observing the movement of these particles you will be able to deduce the structure of the interaction between particles.
Let us  consider the example of the circle moving at constant velocity u in a perpendicular direction to the line. Using pythogoras theorem, we see that

H^2+X^2=R^2.

And because the motion of the circle is uniform, we know that H(t)=u(t_0-t), where t_0 is an arbitrary time origin that we can choose to be zero so that H(t)=-ut. It follows that

X(t)=\sqrt{R^2-u^2t^2}.

Deriving once according to the time, we get the velocity V(t)

V(t)=\frac{dX(t)}{dt}=-{u^2t}/{\sqrt{R^2-u^2t^2}}.

Deriving one more time we now have the acceleration A(t)

A(t)=\frac{dV(t)}{dt}=-{u^2}/{\sqrt{R^2+u^2t^2}}+{u^4t^2}/{\sqrt{R^2+u^2t^2}^3}

which under the same denominator leads to

A(t)=\frac{dV(t)}{dt}=-{u^2R^2}/{X^3}.

since X(t)=L(t)/2 where L(t) is the distance between particles and R=D/2 (where D is the diameter) we conclude that the force that seems to be applied on the particle is

F=A/m=-{2u^2D^2}/{L^3},

where we choose m=1 without loose of generality.

Finally, not knowing that the circle exist, u^2D^2 is, for our tiny physicist, a quantity that he can only measures. It is the constant  that characterise the force. He chooses to call it K:

F=-{2K}/{L^3}.

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