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When can the instantaneous velocity of an object equal the average velocity of an object? I understand that if acceleration is zero, motion is...

As pointed out in the question, with uniform motion (no acceleration) the instantaneous velocity and the average velocity are the same everywhere, as the object has a constant rate of change in regards to its distance traveled.


But there can be times when the instantaneous velocity and average velocity are the same even with varying accelerations. For example, imagine a 300 mile road trip which takes exactly 5 hours. The average speed (velocity if the...

As pointed out in the question, with uniform motion (no acceleration) the instantaneous velocity and the average velocity are the same everywhere, as the object has a constant rate of change in regards to its distance traveled.


But there can be times when the instantaneous velocity and average velocity are the same even with varying accelerations. For example, imagine a 300 mile road trip which takes exactly 5 hours. The average speed (velocity if the trip is all in one direction) is 60 miles per hour. Your speed at any time of the trip could be 60 miles per hour; it might always be 60 miles per hour, or you might speed up and slow down.


This is the premise behind giving speeding tickets to vehicles whose speed has been measured indirectly. Some police utilize markings on the pavement and a flying spotter while others might time a trip between toll stations, etc., but they never measure speed directly. They infer that a vehicle that covers a distance faster than they could have covered it obeying the speed limit must have exceeded the speed limit. Further they know that the vehicle must have reached the average velocity during the trip.


Using ideas from calculus we can show that if the distance function can be modeled by a continuous (no breaks or asymptotes) curve with no corners or cusps over an interval from a to b, then there is a point c in the interval where the velocity (the instantaneous rate of change of the distance at point c) will be equal to the average velocity (the average rate of change over the interval).


You can see this graphically; the curve represents the distance function as a function of time while the line shows the instantaneous velocity for some points:



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