**Problem Statement:**

The potential energy of a particle of mass m is represented in the figure below as a function of its position r. The particle begins its motion from rest at position r_{0}. what will be its speed at position 3r_{0}? If it were released without speed in position 2r_{0}, would it reach 3r_{0}? Why? Determine whether the force acting on the particle is attractive or repulsive for the different segments of the potential energy curve.

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**Solution:**

The mechanical energy of the particle is conserved because, although we do not know the details of how the force acting on it, the fact that there is a potential energy implies that the force is conservative.

We write the equality of the mechanical energy of the particle for the initial (r_{0}) and final (3r_{0}) positions:

Substituting with the potential energy values provided in the figure for these two positions we get:

We can now isolate the speed of the particule at position 3r_{0}:

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If the particle were released at rest in position 2r_{0}, as the potential energy is greater in the final state than in the initial one, the particle could not reach the final state.

We can verify it mathematically by substituting with the values of the potential energy provided in the figure for the positions 2r_{0} and 3r_{0}:

We cannot calculate the speed this time because we would have to calculate the square root of a negative quantity. For the particle to arrive at position 3r_{0} it would need to have a certain speed at position 2r_{0}.

To know if a force is attractive or repulsive for the different segments, it is enough to remember that when a force is conservative, the relation between the force and the potential energy is:

Since the potential energy depends only on one coordinate, it is enough to determine the sign of the slope in each segment of the curve.

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