Projectile arc visualization for Jensen's inequality
Another Jensen's inequality visualization.
Picture the long parabola arc of a thrown ball from the thrower's hand to where it landed yonder. The first sense of average is the midpoint of the distance traveled between the thrower and the landing spot. To see the second sense of average, stand behind the thrower so the projectile's arc is just a vertical line. The second sense of average is the mean height. It's quite intuative how the mean height is lower than the height the ball reached at the mid-point.
For the accelerating case, think of throwing a basketball off a dam to demonstrate the magnus effect. The distance from the dam wall evaluated at the midpoint of the height will be less that the average distance the ball spends away from the dam.
A more difficult analogy for the decelaring case is to throw a ball onto a shelf, thus removing the falling side of the arc. Yet it still holds that at the x midpoint, most of the rising has already happened. The altitude average is lower than the altitude at the x midpoint.
Another way of this is:
- For a decelerating process: everything has already been done by the time you get to the midpoint.
- For an accelerating process: things are just beginning by the time you get to the midpoint; it's all ahead.