Stopping point of a sliding particle.

Question

A particle with given mass > 0 and given coefficient of friction > 0 and given initial downward speed > 0 starts at (0,1) on the graph of y = exp(-x). The coefficient of friction applies only to those points on the graph having a rational horizontal coordinate. Does the particle ever come to a stop?

Answer 1

No.

The total energy of the particle is $E_{tot} = E_{kin} + E_{pot} = \frac{1}{2} mv^2_0 + mg$. In order for friction to stop the particle, the work done by the friction needs to be greater than this. But the work done by the friction is (with some abuse of notation)

$W = \int F \cdot dS = \int \mu_k mg$

As Tom Collinge pointed out, the rational points are a set of measure zero, so this integral is zero. This means that the friction doesn't do ANY work on the particle, and so the particle will slide forever.