In everyday life, animals and humans agree on the Euclidean properties of velocity,
space and time. In particular, this implies that a trajectory can be described by specifying
three numbers, three coordinates (x, y, z) – one for each dimension – as continuous
functions of time t. (Functions are defined in detail on page 826.) This is usually written
as x = x(t) = (x(t), y(t), z(t)). For example, already Galileo found, using stopwatchand ruler, that the height z of any thrown or falling stone changes as
<br />
z(t)=z_0+v_0(t-t_0)-\frac{1}{2}g(t-t_0)^2<br />
where t0 is the time the fall starts, z0 is the initial height, v0 is the initial velocity in the
vertical direction and д = 9.8m/s2 is a constant that is found to be the same, within about
one part in 300, for all falling bodies on all points of the surface of the Earth. Where do
the value 9.8m/s2 and its slight variations come from? A preliminary answer will be
given shortly, but the complete elucidation will occupy us during the larger part of this
hike.
Equation (5) allows us to determine the depth of a well, given the time a stone takes
to reach its bottom.The equation also gives the speed v with which one hits the ground
after jumping from a tree, namely v =\sqrt{2gh} . A height of 3myields a velocity of 27 km/h.
The velocity is thus proportional only to the square root of the height. Does this mean
that one’s strong fear of falling results from an overestimation of its actual effects?
Galileo was the first to state an important result about free fall: the motions in the
horizontal and vertical directions are independent. He showed that the time it takes for
a cannon ball that is shot exactly horizontally to fall is independent of the strength of the
gunpowder, as shown in Figure 41.Many great thinkers did not agree with this statement
even after his death: in 1658 the Academia del Cimento even organized an experiment
to check this assertion, by comparing the flying cannon ball with one that simply fell
vertically. Can you imagine how they checked the simultaneity? Figure 41 also shows
how you can check this at home. In this experiment, whatever the spring load of the
cannon, the two bodies will always collide in mid-air (if the table is high enough), thus
proving the assertion.
In other words, a flying canon ball is not accelerated in the horizontal direction. Its
horizontal motion is simply unchanging. By extending the description of equation (5)
Two expressions for the horizontal coordinates x and y, namely:
x(t) = x0 + vx0(t − t0)
y(t) = y0 + vy0(t − t0), (6)
A complete description for the path followed by thrown stones results.A path of this shape
is called a parabola; it is shown in Figures 18, 41 and 42. (A parabolic Page 45 shape is also used
for light reflectors inside pocket lamps or car headlights. Can you show why?)
Physicists enjoy generalizing the idea of a path. A path is a trace left
in a diagram by a moving object. Depending on what diagram is used, these paths have
different names. Hodographs are used in weather forecasting. Space-time diagrams are
useful to make the theory of relativity accessible. The configuration space is spanned by
the coordinates of all particles of a system. For many particles, it is has a high number of
dimensions. It plays an important role in self-organization.The difference between chaos
and order can be described as a difference in the properties of paths in configuration
space. The phase space diagram is also called state space diagram. It plays an essential
role in thermodynamics.
