An object of negligible size slides on a horizontal plane and its initial speed is \(v_{0}=4 \mathrm{~m} / \mathrm{s}\). The surface of the plane has increasing roughness and the corresponding friction force can be described by a linearly increasing dynamic friction coefficient, \(\mu_{d}=\mu_{0} \cdot x\) where \(x\) (in meters) is the position along the direction of motion and \(\mu_{0}=0.03 \mathrm{~m}^{-1}\) is a constant. The coordinate point \(x=0\) corresponds to the initial position.

1. Write the expression of the work that the frictional force does as a function of \(x\).

2. Calculate the distance \(D\) traveled by the object to its stop.

3. Write the differential dynamic equation and the equation of motion.

4. Determine the time \(t^{*}\) the object takes to stop.

5. The obtained equation of motion is clearly not valid for \(t>t^{*}\). What physical condition must be mathematically expressed for the law to be valid for every value of \(t\) ?