Average Velocity: Initial & Final Velocities vs. Mean Value Theorem

AI Thread Summary
The discussion focuses on deriving the formula for average velocity as the sum of initial and final velocities divided by two, with uncertainty about a specific step in the derivation. An alternative approach is proposed, suggesting that calculating velocity at every instant and summing these values leads to the integral of the velocity function, which aligns with the Mean Value Theorem for Integrals. The relationship between average velocity and displacement over time is emphasized, reinforcing that average velocity is the vector difference in displacement divided by the time interval. The validity of both methods for deriving average velocity is questioned. Overall, the conversation explores different mathematical approaches to understanding average velocity.
Ali Asadullah
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In the photo attached, i have tried to derive the formula that average velocty is equal to the sum of initial and final velocities divided by two. But i am not sure about one step encircled in the photo. I don't know whether encircled step is right or wrong. I have another idea of deriving this formula which is similar to this but skips this step.
I assumed that we calculate velocities after each one second, but if we suppose that we somehow calculate velocity after each instant then sum of velocities will gives us the integral of velocity function and dividing it by "n" will give us average velocity according to Mean Value Theorem for Integrals.
My question is whether my second appraoch is right?
 

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Average velocity is equal to the vector difference in displacement divided by the time interval
 

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