The equations below describe the motion of a falling body, assuming that the acceleration due to gravity is a constant,
g (in which case Newton's law of gravitation simplifies to
F =
mg where m is the mass of the earth). This assumption is reasonable for objects falling to earth over the relatively short vertical distances of our everyday experience, but is very much untrue over larger distances (such as spacecraft trajectories).
Galileo was the first to demonstrate and then formulate these equations. He used a
ramp to study rolling balls, the ramp slowing the acceleration enough to measure the time taken for the ball to roll a known distance. He measured elapsed time with a
water clock, using an "extremely accurate balance" to measure the amount of water.
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The equations ignore air resistance, which has a dramatic effect on objects falling an appreciable distance in air, causing them to quickly approach a
terminal velocity. For example, a person jumping headfirst from an airplane will never exceed a speed of about 200 mph due to air resistance. The effect of air resistance varies enormously depending on the size and geometry of the falling object – for example, the equations are hopelessly wrong for a feather, which has a low mass but offers a large resistance to the air. (In the absence of an atmosphere all objects fall at the same rate, as astronaut
David Scott demonstrated by dropping a hammer and a feather on the surface of the
Moon.)
The equations also ignore the rotation of the Earth, failing to describe the
Coriolis effect for example. Nevertheless, they are usually accurate enough for dense and compact objects falling over heights not exceeding the tallest man-made structures.
Near the surface of the Earth, use
g = 9.8 m/s2 (metres per second per second), approximately. For other planets, multiply
g by the appropriate
scaling factor. It is essential to use consistent units for
g,
d,
t and
v. Assuming
SI units,
g is measured in metres per second per second, so
d must be measured in metres,
t in seconds and
v in metres per second. To convert metres per second to kilometres per hour (km/h) multiply by 3.6. In all cases the body is assumed to start from rest.
Distance
d travelled by an object falling for time
t:
Time
t taken for an object to fall distance
d:
Instantaneous velocity
vi of a falling object after elapsed time
t:
Instantaneous velocity
vi of a falling object that has travelled distance
d:
Average velocity
va of an object that has been falling for time
t (averaged over time):
Average velocity
va of a falling object that has travelled distance
d (averaged over time):
