Galileo's Experiment: Uniformly Accelerated Motion

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Galileo's experiment demonstrated that the distance covered by a body in free fall is proportional to the square of time, indicating uniformly accelerated motion, as acceleration is defined as the second derivative of distance over time. The discussion highlights that in non-uniformly accelerated motion, distance does not follow this time-squared relationship. Galileo concluded that the acceleration is constant for all bodies by conducting multiple experiments with various objects, consistently observing the same acceleration value regardless of mass. This empirical evidence supports the idea that acceleration due to gravity is uniform across different objects. Overall, Galileo's findings laid the groundwork for understanding uniform acceleration in physics.
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1. how can the experimental result that the distance covered by a body during free fall is directly proportional to the square of time imply that the motion is uniformly accelerated? (of course the result agrees with one of a property of uni. accelerated motion but can we show that in nonuniformly accelerated motion the distance is not proportional to time squared?)

2. galileo showed that for a free fall is uniformly accelerated. but how did he show that the acceleration which is constant for a body is same for all other bodies? (i know the statement is implied from the Newtons second law. but how did galielo conclude it?)
 
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1. This one's easy! d(t) = kt2, and acceleration is defined as the second derivative of d(t), which is a constant.

2. I'm not sure he did. I know he showed you get the same acceleration independent of the mass.
 
2) Very simple: He did the experiment many times using many diffrerent objects and always got the same value for the acceleration.
 

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