Elapsed time is the amount of time that has been spent on an action. It does not necessarily have to start at t=0 For example, a ball travels for 4s at 20m/s, at 4s, the ball begins parabolic motion and continues with the motion until it strikes the ground, at 4s how long did it take for the ball to reach the ground. Although it is not starting at t=0 it is basically treated as such because the time before the ball begins to decline is used for nothing more than to find initial displacement.
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