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Velocity is defined to be the rate of change of position i.e how fast an object moves irrespective of from where and when. For example- An object moves a to b covering a distance of 5km and b to c covering a distance of 5 km. And a to c is 2.5km. It take 10 hr to reach the final destination. So the displacement is 2.5km. And the Velocity is (2.5/10)km/hr

. Now suppose the object had speed up while traveling a to b and now it took 9 hrs to cover the same distance. So the Velocity now is (2.5/9)km/hr.

So you see the Velocity is constant throughout the path in case I i.e (2.5/10)km/hr And in case 2 also it is constant throughout the path i.e (2.5/9)km/hr.[Though the direction does change but magnitude remains constant]

So the physical quantity which is accelerating is speed but acceleration is defined as the rate of change of Velocity.

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You confuse average velocity with velocity at a moment. Only if you have no acceleration you can calculate the velocity by distance divided by time. This gives only the average velocity. if you accelerate the velocity is not constant. the Definition of velocity is $v=\frac{ds}{dt}$ not $\frac{\Delta s}{\Delta t}$ if you are not familiar with derivatives you can think of velocity at a very short time if you make $\Delta s$ very small.

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