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In physics, acceleration is how quickly one's speed (more correctly, velocity) changes. Since velocity is usually given in metres per second, and we ask how quickly that changes, we have to say that the rate of change of velocity is expressed as (metres/second) per second, or metres per second squared.
In symbols, $ms^{2}.$
Sir Isaac Newton expressed three laws of motion, the second of which says that the acceleration of an object is proportional to the force being applied, and inversely proportional to its mass. For a given mass, applying twice the force results in twice the acceleration, while for a given constant force, an object that is twice as heavy will accelerate half as fast.
Why then do heavier objects not fall more slowly?
Because the force due to gravity is proportional to the mass of the object. Thus for an object that is twice as heavy the force is twice as great, and so these effects exactly balance, and the acceleration due to gravity is the same for all objects (ignoring such effects as air resistance).
See also Galileo.
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