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Motion: Why we need acceleration??

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Hi guys!!
While studying motion of any object we need displacement (say, x) of the object(to locate the object from some reference)to describe it's motion at any time instant.
Next to compare the motion (like how fast the object moves)we need to determine velocity/speed of the object.
But why do we need acceleration to describe the motion? Aren't displacement & velocity enough? Is it necessary that we need to know the acceleration of the object to describe the complete motion?

Now if at all we need acceleration(dv/dt) then why don't we consider jerk,snap etc.(the subsequent derivatives of displacement) to describe an object's motion?
Note: I am talking about any general motion and not necessarily rectilinear translation.
Thanks.

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