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If I have a sphere rolling on a table with an initial velocity and then it falls off the edge of the table, from this point on it is taken as if my initial velocity were zero, or it starts from the initial velocity that I had before in the table?
The initial velocity after leaving the table is the same as it was before leaving the table (nothing is present to produce discontinuties in the velocity).
Before leaving the table the vertical velocity was zero and is zero when it leaves the table.
The horizontal velocity is constant with no forces present to change the velocity.
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