What if the initial position of the pumpkin is f(5)? does it make any difference?

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You have to calculate the average velocity accordingly. you can find the average velocity from t(5) to t(6) using his methods, or from t(5) to t(7). In what way are you asking if it makes a difference?  If it starts in a different position, then the path of the pumpkin is different. Just use or construct a table and use these methods for the different time intervals.