We know average velocity in its strictest sense, means total displacement over total time taken: $$\frac{X_f-X_i}{T_f-T_i}$$
There's a special case, when a body is moving in a straight line with a constant acceleration. Of course since its acceleration is constant, it has to be a rectilinear motion. In this case average velocity (over a time interval) is simply $$V_{avg}=\frac{V_1+V_2}{2} \tag 1$$
It can be proved easily, using the equations of motion. The important point is, this formula works only when a body is moving with a "constant acceleration". As per my teachers and the books I have.
The problem is, why I'm actually putting this post up, there's another case. If the body moves with a velocity $V_1$ for a time interval $t$, and then it moves with a velocity $V_2$ for the same amount of time $t$. In other words, the body traveling with a velocity $V_1$, takes time $t$ to go from a point $A$ to another point $B$, and then it goes from point $B$ to another point $C$ and again, it takes time $t$, traveling with a velocity $V_2$.
In this case, when time intervals are equal, we calculate average velocity (or average speed) by taking the arithmetic mean of individual velocities, and this formula (as per my teachers and the books I have). So, in this case : $$V_{avg}=\frac{V_1+V_2}{2}\tag 2$$
But equation (1) and equation (2) are completely identical. Isn't that strange? And odd? Because equation (1) should be valid 'only' when the acceleration is constant. But in the second case, acceleration of the body is not constant during the course of its motion. It's acceleration is zero as it goes from $A$ to $B$, then its acceleration changes as it velocity changes from $V_1$ to $V_2$, and then its acceleration is constant from $B$ to $C$. Even though its acceleration is non-constant, the formula for finding out its average velocity is exactly the same, as in the first case.
Please explain what's going on here. Because the formula $V_{avg}=\frac{V_1+V_2}{2}$ should be valid if and only if acceleration is uniform. (As per my books and my teachers). Thanks