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But here the graph shows that the potential energy is minimum during collision.

But it doesn't. Keep in mind that $r$ becomes smaller as the balls get closer together and then becomes larger as they move apart. The horizontal axis is not time. So the potential energy actually increases to a maximum at the minimum separation, and then it goes back down as the balls separate.

As for the other points, it makes sense that you want $V$ to increase as $r$ decreases. This is because $$F=-\frac{\text dV}{\text dt}$$ The inverse relationship between $V$ and $r$ means that the force is pushing the balls in contact towards being farther apart.

None of the other answer choices give you the correct force behavior (check for yourself).

To answer your title question, this does not always have to be the case. But for your system we don't want any attraction between the balls, so we do want this behavior here. It doesn't necessarily have to be inversely proportional, but it does need to monotonically decrease as $r$ increases (i.e. an inverse relationship).

But here the graph shows that the potential energy is minimum during collision.

But it doesn't. Keep in mind that $r$ becomes smaller as the balls get closer together and then becomes larger as they move apart. The horizontal axis is not time. So the potential energy actually increases to a maximum at the minimum separation, and then it goes back down as the balls separate.

As for the other points, it makes sense that you want $V$ to increase as $r$ decreases. This is because $$F=-\frac{\text dV}{\text dr}$$ The inverse relationship between $V$ and $r$ means that the force is pushing the balls in contact towards being farther apart.

None of the other answer choices give you the correct force behavior (check for yourself).

To answer your title question, this does not always have to be the case. But for your system we don't want any attraction between the balls, so we do want this behavior here. It doesn't necessarily have to be inversely proportional, but it does need to monotonically decrease as $r$ increases (i.e. an inverse relationship).

But here the graph shows that the potential energy is minimum during collision.

But it doesn't. Keep in mind that $r$ becomes smaller as the balls get closer together and then becomes larger as they move apart. The horizontal axis is not time. So the potential energy actually increases to a maximum at the minimum separation, and then it goes back down as the balls separate.

As for the other points, it makes sense that you want $V$ to increase as $r$ decreases. This is because $$F=-\frac{\text dV}{\text dt}$$ The inverse relationship between $V$ and $r$ means that the force is pushing the balls in contact towards being farther apart.

None of the other answer choices give you the correct force behavior (check for yourself).

To answer your title question, this does not always have to be the case. But for your system we don't want any attraction between the balls, so we do want this behavior here. It doesn't necessarily have to be inversely proportional, but it does need to monotonically decrease as $r$ increases.

But here the graph shows that the potential energy is minimum during collision.

But it doesn't. Keep in mind that $r$ becomes smaller as the balls get closer together and then becomes larger as they move apart. The horizontal axis is not time. So the potential energy actually increases to a maximum at the minimum separation, and then it goes back down as the balls separate.

As for the other points, it makes sense that you want $V$ to increase as $r$ decreases. This is because $$F=-\frac{\text dV}{\text dt}$$ The inverse relationship between $V$ and $r$ means that the force is pushing the balls in contact towards being farther apart.

None of the other answer choices give you the correct force behavior (check for yourself).

To answer your title question, this does not always have to be the case. But for your system we don't want any attraction between the balls, so we do want this behavior here. It doesn't necessarily have to be inversely proportional, but it does need to monotonically decrease as $r$ increases (i.e. an inverse relationship).

But here the graph shows that the potential energy is minimum during collision.

But it doesn't. Keep in mind that $r$ becomes smaller as the balls get closer together and then becomes larger as they move apart. The horizontal axis is not time. So the potential energy actually increases to a maximum at the minimum separation, and then it goes back down as the balls separate.

As for the other points, it makes sense that you want $V$ to increase as $r$ decreases. This is because $$F=-\frac{\text dV}{\text dt}$$ The inverse relationship between $V$ and $r$ means that the force is pushing the balls in contact towards being farther apart.

None of the other answer choices give you the correct force behavior (check for yourself).

To answer your title question, this does not always have to be the case. But for your system we don't want any attraction between the balls, so we do want this behavior here. However note that it not "inversely proportional" to $r$. What we want is for $V$ to decrease as $r$ increases. This is the "inverse" relationship they refer to I suppose.

But here the graph shows that the potential energy is minimum during collision.

But it doesn't. Keep in mind that $r$ becomes smaller as the balls get closer together and then becomes larger as they move apart. The horizontal axis is not time. So the potential energy actually increases to a maximum at the minimum separation, and then it goes back down as the balls separate.

As for the other points, it makes sense that you want $V$ to increase as $r$ decreases. This is because $$F=-\frac{\text dV}{\text dt}$$ The inverse relationship between $V$ and $r$ means that the force is pushing the balls in contact towards being farther apart.

None of the other answer choices give you the correct force behavior (check for yourself).

To answer your title question, this does not always have to be the case. But for your system we don't want any attraction between the balls, so we do want this behavior here. It doesn't necessarily have to be inversely proportional, but it does need to monotonically decrease as $r$ increases.