If you drop an object on earth, gravity will increase its speed by 32
feet per second every second. Thus after one second it will fall at a
speed of 32 feet per second, and after two seconds its speed will be
64 feet per second. We say that gravity is accelerating the object at
**32 feet per second per second**, or, more concisely, **32 feet
per second squared.** The particular number 32 is due to the mass of
the earth and its radius. It is different on other bodies in the
solar system, as indicated in the following table:

Earth | Moon | Sun | Mars | Venus | Mercury | |

32 | 5 | 892 | 12 | 29 | 12 | feet/second |

For generality and clarity let's say that the speed of a falling object increases by (where stands for

For example, if you toss a rock upwards at an initial velocity of 20 feet per second, and you release it at a height of 5 feet, then its height is described by

Suppose now that in addition to tossing that rock upwards you also
toss it *forward*, let's say at a (horizontal) speed of 25 feet per second. You
wonder how far that rock will fly. The horizontal and the vertical
components of the rock's motion are independent, and since no
horizontal force acts on the rock (ignoring air resistance) its
forward velocity is constant. Thus the rock flies a distance of
feet before it hits the ground.

Figuring out ** how high** the rock will fly is a little more tricky.
You will learn effective ways of answering questions like this in
Calculus. At this stage the easiest way to compute the maximum height
is to complete the square. We have

To determine the maximum you could also first ask at what time the vertical velocity is zero, and then evaluate the height at that time.

To determine the time at which an object is at a particular height you solve a quadratic equation obtained from the height formula. In particular, to find the time an object hits the ground you set the height to zero. To find the maximum height complete the square or ask when the vertical velocity is zero. To find a distance first find the time at which an object hits the ground, and then multiply the horizontal velocity with that time.

The discussion on this page is fundamentally flawed by the fact that it ignores air resistance. It also ignores the fact that the earth is spherical rather than planar. The formulas on this page are approximately valid for small speeds and heights, and for the purposes of the exercises in this class we will ignore air resistance and the shape of the earth.