I often see the formula:
$$
\overline v = \frac {v + v_0}{2}
$$
in physics textbooks, to describe the average velocity for an object moving with constant acceleration. This formula is often substituted into the $x = x_0 + \overline v t$ formula in order to derive the $x = x_0 + v_0t + 1/2at^2$ formula we are so familiar with.
With a calculus background we don't even need this formula. We just keep integrating from a = constant until we get the same formula. But if you don't have a calculus background I guess using this first equation is a necessary stepping stone on the way to deriving the formula.
That the average velocity is the mid-point between the initial and final velocities makes intuitive sense I guess, but I want to be able to demonstrate to students that this is true in a more rigorous way. The only problem is that the only way I can see to proving that this formula is indeed true is to use calculus!
In particular, I would use the knowledge that the area under the v/t curve is the displacement. Drawing a simple v/t graph that is linear (constant acceleration) I could show geometrically that the area under this curve is the same formula.
And there's the problem. The area under the curve bit of knowledge implies a knowledge of calculus.
Is there any other way to demonstrate to a non-calculus student that it is true? I've never seen any textbook attempt to justify this formula. It just seems to pop out of nowhere without any rigorous proof.
Best Answer
How about this:
$v(t)$ is a line. This should be an OK assumption. So it's of the form $mt+b$. If you consider the interval $t_1$ through $t_2$ and draw a horizontal line of height $\frac{1}{2}(v(t_1)+v(t_2))=m \frac{1}{2}(t_1+t_2)+b$, you'll see that exactly half of the velocity line lies above the horizontal line, and exactly half lies below. Therefore that must be the average.
If the "exactly half above, exactly half below" argument doesn't hold (for discrete numbers that's the median, not the mode), discretize the line by $N$ points and prove the average of that is always $\frac{1}{2}(v(t_1)+v(t_2))$. I.e., prove
$$\frac{1}{N+1}\sum_{n=0}^N v(t_n)= \frac{1}{2}(v_0+v_f)$$