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Time square by itself doesn't seem to make much sense, right?

You must consider that acceleration is NOT the same thing as velocity. Velocity (and speed) is expressed (for example) in meter/second. An acceleration is a change of velocity per time unit, so logically the units are (meters/second) / second. This is the same as meters/second2, and that is the way it is usually written.

Answer

Acceleration is the rate of change of velocity. In other words, it is velocity divided by time. As velocity (in SI) is expressed in metres per second, acceleration must be measured in 'metres per second per second'. In fact, this is the correct way of saying the unit of measurement for acceleration, NOT 'metre per second squared'. The symbol for metres per second per second is 'm/s/s' which, of course, is exactly the same as m/s2 (where 2 is a superscript).

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