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There are two possible scenarios:

1. The fast accelerating person is in front.
2. The slowly accelerating person in in front.

In case 2 there is no tension (the rope is not taut).*

In case 1, if the rope is rigid it will break. Otherwise the different accelerations are not possible.

Alternatively, if the rope is elastic, then elastic tension (spring force) will increase since the two people will separate more and more, corresponding to elongating the rope.

^{* Since the rope is not massless, gravity might have an influence causing some tension within the rope depending on how it is carried and how it is hanging in its non-taut state. Finding the tension distribution depends highly on the exact shape with which it is hanging but can possibly be compared to a cable hanging between pillars of a suspension bridge.}

There are two possible scenarios:

1. The fast accelerating person is in front.
2. The slowly accelerating person in in front.

In case 2 there is no tension (the rope is not taut).*

In case 1, if the rope is rigid it will break. Otherwise the different accelerations are not possible.

Alternatively, if the rope is elastic, then elastic tension (spring force) will increase since the two people will separate more and more, corresponding to elongating the rope.

^{* If the rope is not massless, gravity might have an influence causing some tension within the rope depending on how it is carried and how it is hanging in its non-taut state. Finding the tension distribution depends highly on the exact shape with which it is hanging but can possibly be compared to a cable hanging between pillars of a suspension bridge.}

There are two possible scenarios:

1. The fast accelerating person is in front.
2. The slowly accelerating person in in front.

In case 2 there is no tension (the rope is not taut).

In case 1, if the rope is rigid it will break. Otherwise the different accelerations are not possible.

Alternatively, if the rope is elastic, then elastic tension (spring force) will increase since the two people will separate more and more, corresponding to elongating the rope.

There are two possible scenarios:

1. The fast accelerating person is in front.
2. The slowly accelerating person in in front.

In case 2 there is no tension (the rope is not taut).*

In case 1, if the rope is rigid it will break. Otherwise the different accelerations are not possible.

Alternatively, if the rope is elastic, then elastic tension (spring force) will increase since the two people will separate more and more, corresponding to elongating the rope.

^{* Since the rope is not massless, gravity might have an influence causing some tension within the rope depending on how it is carried and how it is hanging in its non-taut state. Finding the tension distribution depends highly on the exact shape with which it is hanging but can possibly be compared to a cable hanging between pillars of a suspension bridge.}

There are two possible scenarios:

1. The faster accelerating person is in front.
2. The slower accelerating person in in front.

In case 2 there is no tension (the rope is not taut).

In case 1, if the rooe rigid it will break. Otherwise the different accelerations are not possible. Alternatively, if the rope is elastic, then elastic tension (spring force) will grow since the two people will get separated more and more, elongating the rope.

There are two possible scenarios:

1. The fast accelerating person is in front.
2. The slowly accelerating person in in front.

In case 2 there is no tension (the rope is not taut).

In case 1, if the rope is rigid it will break. Otherwise the different accelerations are not possible. 

Alternatively, if the rope is elastic, then elastic tension (spring force) will increase since the two people will separate more and more, corresponding to elongating the rope.