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Suppose there is the following situation:

Blocks $A$ and $B$, with masses $m_A$ and $m_B$, are connected by a light spring on a horizontal, frictionless table. When block $A$ has acceleration $a_A$, then block $B$ has, by Newton’s second and third laws, acceleration $-a_A\frac{m_A}{m_B}$.

What does it mean for block $B$ to have a negative acceleration? Since the blocks are connected by a spring, does not block $B$ move in the same direction as block $A$?

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3 Answers

What does it mean for block B to have a negative acceleration? Since the blocks are connected by a spring, does not block B move in the same direction as block A?

Since they are connected by a spring, Block B does not move in the same direction as block A always. Actually, this situation can be understood using center of mass.

In the situation given, the masses will move in the positive x direction while oscillating back and forth along the spring. Also, it isn't necessary that the Block B "moves" backward, although here it seems it does. It's just the acceleration which is negative, which is justifies if they are oscillating.

Another case might be that there are other forces acting on Block B. What you wrote is just the acceleration of Block B due to Block A.

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Negative acceleration relative to direction essentially means the mass is decelerating i.e. slowing down motion in that direction. In this system, we essentially model that the spring will oscillate as the system moves, which means there will be points when B is moving away from A in the centre of mass frame.

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Acceleration is a vector, so $-a_A$ denotes an acceleration with the same magnitude as $a_A$, but in the opposite direction.

As for the second part, imagine that I place the two blocks down at rest, but far apart such that the spring is stretched beyond it's natural length. The spring will apply a force on each block towards the midpoint, causing them to move towards each other- opposite directions.

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