CONTACT OF ELASTIC SOLIDS
161
the other hand, dJ is equal to the element of time dt, multi-
plied by the pressure which during that time acts between the
bodies. This is ka¹, where k₂ is a constant to be determined
from what precedes, which constant depends only on the form
of the surfaces and the elastic properties quite close to the
point of impact. Hence dJ kalt and da' - k₂ka¹dt;
integrating, and denoting by a', the value of a' just before
impact, we find
a'?
energy.
=
a²+3a² = 0,
=
which equation expresses the principle of the conservation of
When the bodies approach as closely as possible a'
vanishes; if am denote the corresponding value of a, then
5a), and the simultaneous maximum
am
=
4h₂h₂
pressure is
Pm=ka. From this we at once obtain the dimensions of
the surface of impact.
In order to deduce the variation of the phenomenon with
the time, we integrate again and obtain
=
In
da
-
The upper limit is so chosen that t= O at the instant of nearest
approach. For each value of the lower limit a, the double
sign of the radical gives two equal positive and negative values
of t. Hence a is an even and a' an odd function of t; im-
mediately after impact the points of impact separate along
the normal with the same relative velocity with which they
approached each other before impact. And the same tran-
scendental function which represents the variation of a' between
its initial and final values, also represents the variations of all
the component velocities from their initial to their final values.
In the first place, the bodies touch when a = 0; they
separate when a again = 0. Hence the duration of contact,
that is the time of impact, is
M. P.
Ꭲ
am
da
I=2] Ves - thika!
―
M
5
25
a
ทะ
=
2n
=
2n
16a