Imagine that you find yourself to be a guest at an anniversary celebration of a federation colony on some strange cratered planetoid. The colonists are staging a high-tech celebration. They fire a cannonball into the night sky. In its wake, the cannonball leaves behind a luminous rainbow trail made up of nondispersing pigments. …

You are in the audience … watching from the surface of the planetoid right below the rainbow. As the cannonball traces its perfect parabola, you take pictures of it every second with your camera, which has a built-in range finder. You find that the vertical height of the cannonball at 0 seconds is 20 meters above you; at 1 second, 48 meters; at 2 seconds, 62 meters; at 3 seconds, 62 meters; at 4 seconds, 48 meters; and at 5 seconds, 20 meters. You have to answer three questions:

1) What is the gravitational acceleration on the surface of this planetoid?
2) What is the vertical velocity of the cannonball?
3) What is the vertical height of the cannon?

Here is the technique:
You line up the numbers in a row. Call this sequence h for height. Below this, you make another sequence (call it v for velocity), where every number is the difference of the two adjacent numbers in the row above (the right number minus the left). Below this row, make a third sequence in the same way, calling it a for acceleration. Here’s what it’ll look like when you are done.

h: 20 48 62 62 48 20
v: 28 14 0 -14 -28
a: -14 -14 -14 -14

If you are at all familiar with calculus, you may know that the vertical velocity is the first derivative of the height, and the acceleration is the second. If you didn’t, don’t worry about it: the simple procedure for second graders that we just did will give us the answers. They are as follows:

1. The gravitational acceleration (g) at the planetoid surface is the number in the last row. This comes out to be -14 meters/sec/sec. … (The minus sign indicates that it is downward in direction). For future reference, remember half this number (-7).
2. The vertical velocity of the cannonball is the first number in the second row minus half the g that we just determined. It is therefore 28 – (-7) = 35 meters/sec.
3. The vertical height of the cannon (obviously) is the first number in the first row. That’s 20 meters.

Now look what else we’ve accomplished! Given a sequence of numbers, we have, easy as pie, found the polynomial formula for it: it is the sum of the three numbers above: -7x2 + 35x +20. You can substitute the numbers 0 through 5 for x and confirm that you get the original series.