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- TL;DR Summary
- How to remove the singularity from the derivative of s^2=x^2+y^2

Setting: a plane with the standard Cartesian coordinate system. A particle is constrained to the x axis, with position x and moving at speed x dot. Another particle is constrained to the y axis, with position y and moving at speed y dot. The distance between them at any moment is s. It is easy to show that this distance changes at rate

[itex] \dot {s} = (1/s)(x \dot{x} + y \dot{y} ) [/itex]

which seems fine except for the singularity when both particles are simultaneously at the origin. In that special case, with geometric reasoning one can arrive at

[itex] \dot{s}^2 = \dot {x} ^2 + \dot {y} ^2 [/itex]

But how do I remove the singularity from my first s dot equation to arrive at an equivalent form that doesn't contain the singularity (and, I assume, simplifies to the second equation in the special x=0, y=0 case)?

[itex] \dot {s} = (1/s)(x \dot{x} + y \dot{y} ) [/itex]

which seems fine except for the singularity when both particles are simultaneously at the origin. In that special case, with geometric reasoning one can arrive at

[itex] \dot{s}^2 = \dot {x} ^2 + \dot {y} ^2 [/itex]

But how do I remove the singularity from my first s dot equation to arrive at an equivalent form that doesn't contain the singularity (and, I assume, simplifies to the second equation in the special x=0, y=0 case)?