__The
Downward Motion of Particles in a Uniform Gravitational Field
__

__1.
Introduction.
__

In
this article the downward motion of a particle with mass, m, in a uniform
gravitational field, g, is discussed.

__2.2.
Solution.
__

Let us begin
by considering the downward velocity and acceleration of a particle in a uniform
gravitational field, g. The acceleration can of course not be constant since v
must always be less than c. But it is given that **g**,
the gravitational field, *is* constant.
And the particle’s gravitational mass at any time t is a function of its
speed:

m(t)=g(t)m_{0}.
(2.1)

The
gravitational force law thus indicates that the gravitational force points
“downward”:

**F**_{grav}(t)=g(t)m_{0}**g**.
(2.2)

By Newton 3,
however, the mechanical force must be

**F**_{mech}(t)=g(t)^{3}m_{0}**a**(t)**.**
(2.3)

Assuming
there can be only one force acting on the particle at any given time, it must be
that

g(t)m_{0}**g=** g(t)^{3}m_{0}**a(t)**.
(2.4)

This being
the case,

**a(t)**=g(t)^{-2}**g,**
(2.4)

and

v(t)=(1-a(t)/g)^{1/2}c.
(2.5)

It is
noteworthy that v and a are independent of m_{0}; all particles with the
stated initial conditions have the same velocity and acceleration after a time t
has elapsed.

The
analytical solutions for v(t) and a(t) are mathematically challenging and
suggest little about the particle’s motion. The values can, however, be
numerically computed. Starting with initial conditions of v(0)=0 and
a(0)=-c/sec, Fig. 1 plots v(t) vs. t and Fig. 2 plots a(t) vs. t. Note how v
asymptotically approaches c, and how a asymptotically approaches zero. A program
for the numerical solution is provided in Appendix A.

Figure
1

v(t)
vs. t

Figure
2

__Appendix
A
__

#COMPILE EXE

#DIM ALL

FUNCTION PBMAIN () AS LONG

DIM c AS DOUBLE

c=3e8

DIM g AS DOUBLE

g=-3e8

DIM n AS LONG

n=2500000

DIM t(n) AS DOUBLE

DIM dt AS DOUBLE

dt=1e-6

DIM v(n) AS DOUBLE

DIM a(n) AS DOUBLE

v(0)=0

a(0)=g

DIM index AS LONG

FOR index=1 TO n-1

t(index)=index*dt

v(index)=v(index-1)+a(index-1)*dt

a(index)=(1-v(index)^2/c^2)*g

NEXT

OPEN "c:\\users\Marjorie Dixon\Documents\speeds.dat" FOR OUTPUT AS #1

OPEN "c:\\users\Marjorie Dixon\Documents\accelerations.dat" FOR OUTPUT AS #2

FOR index=0 TO n-1

IF index MOD 1000=0 THEN

WRITE #1, t(index),v(index)

END IF

NEXT

FOR index=0 TO n-1

IF index MOD 1000=0 THEN

WRITE #2, t(index), a(index)

END IF

NEXT

CLOSE #1

CLOSE #2

MSGBOX("ready for plotting")

END FUNCTION