Distance = 16_t_²,
we learn that a body falls through 64' in 2 seconds; and as we know
that it falls 16' in the first second, it must fall 48' in the next
second. Let us examine this. After falling for one second, the body
acquires a certain velocity, and with that velocity it commences the
next second. Now, according to what we have just seen, gravity will act
during the next second quite independently of whatever velocity the
body may have previously had. Hence in the second second gravity pulls
the body down 16', but the body moves altogether through 48'; therefore
it must move through 32' in consequence of the velocity which has been
impressed upon it by gravity during the first second. We learn by this
that when gravity acts for a second, it produces a velocity such that,
if the body be conceived to move uniformly with the velocity acquired,
the body would in one second move over 32'.
517. In three seconds the body falls 144', therefore in the third
second it must have fallen
144' - 64' = 80';
but of this 80' only 16' could be due to the action of gravity
impressed during that second; the rest,
80' - 16' = 64',
is due to the velocity with which the body commenced the third second.
518. We see therefore that after the lapse of two seconds gravity
has communicated to the body a velocity of 64' per second; we should
similarly find, that at the end of the third second, the body has a
velocity of 96', and in general at the end of _t_ seconds a velocity
of 32_t_. Thus we illustrate the remarkable law that _the velocity
developed by gravity is proportional to the time_.
519. This law points out that the most suitable way of measuring
gravity is by the velocity acquired by a falling body at the end of
one second. Hence we are accustomed to say that _g_ (as gravity is
generally designated) is 32. We shall afterwards show in the lecture on
the pendulum (XVIII.) how the value of _g_ can be obtained accurately.
From the two equations, _v_ = 32_t_ and _s_ = 16_t_² it is easy to
infer another very well known formula, namely, _v_² = 64_s_.