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If we set the following system in motion by exerting a force on one of the objects, the acceleration of both blocks will be the same.

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I'm wondering why the acceleration of both blocks will be the same? I understand that we can draw this conclusion from doing an experiment and looking at the result, but I'm wondering if there's an explanation from the theoretical perspective, such as one that addresses the effect the applied force has on the objects at the atomic level.

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Each block consists of many atoms. You could also ask: Why do those atoms have the same acceleration?

The answer is the strong atomic and chemical bonds. The force you apply to the first particle will propagate to the next particle via these bonds. For a perfectly rigid material (the ideal, theoretical situation) with no elastic behaviour (no delayed force propagation), the force exerted on the first particle will immediately be felt by the last particle.

The taut, perfectly inelastic rope that connects two blocks is no different in this regard.

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Because the string keeps the two objects the same distance apart. If one moves a short distance $\Delta x_1$ in a short time $\Delta t$, the velocity is $v_1 = \Delta x_1 /\Delta t$. The second object moves the same $\Delta x_1$ in the same $\Delta t$, so it has the same velocity.

A short time later, the objects are moving faster. They both move a distance $\Delta x_2$ in time $\Delta t$, and the velocity of both is $v_2 = \Delta x_2 /\Delta t$.

The acceleration of both can be calculated from how their velocities have changed. $a = \Delta v / \Delta t = (v_2 - v_1)/\Delta t$. It is the same for both.

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