__Escape
Velocities in Single-Valued g Fields
__

**Abstract.
**A**
r**elativistic mass, attached to an ideal spring and in a constant, opposing gravitational field, is modeled. Three cases are considered: (1) The
gravitational force is such that the mass is captured by the spring, (2) The
gravitational force is such that the mass escapes, stretching the spring to
infinity, (3) The gravitational force is such
that the computer calculates a speed greater than c, and a fatal runtime error
occurs..

**Discussion.**
In most of the literature, the escape velocity is calculated for a free (no
opposing spring) mass in an *inverse square* g field. The mass may or may
not be captured, depending on its initial velocity and the ambient field
strength. One might at first surmise that, in a *uniform*
field a mass *on an opposing spring* will *always* be
captured. For one might reason that the spring force grows with increasing
penetration of the mass into the field and, assuming the spring force magnitude
eventually exceeds the gravitational force on the mass, the mass must eventually
reverse direction and hence be captured.

_{0}g. In
the case of mass and gravity,
however, the gravitational force is *not *constant;
it equals _{0}g.
And there is a threshold initial velocity and g field strength combination at which f_{grav} is *always*
greater than f_{spring}, and the mass escapes!

It is also worth
noting that a charge in a constant *electric* field would *always* be
captured, since the electric force would be a speed-independent qE, and
eventually the spring force would exceed the electric force.

The Python program
that computes each case is provided in Appendix A.

__Appendix
A
__

#Python
3.3.4 (v3.3.4:7ff62415e426, Feb 10 2014, 18:12:08) [MSC v.1600 32 bit (Intel)]
on win32

#Type
"copyright", "credits" or "license()" for more
information.

#constant
gravitational field, attached
to an ideal spring of spring constant k.

#The
mass' speed, at x=0 and t=0, is relativistic.

#The
goal of the modeling program is to demonstrate that the mass is captured

#in
the first field, but escapes at more powerful fields.

#For
g>=3e13, v actually exceeds c in the program

#and
a 'sqrt of negative number' runtime error occurs.

import
math

g=3e7
#Choose ambient field strength by uncommenting.

#g=3e8

#g=3e13

n=1000000
#Loop counter

c=3e8
#Speed of light

v=2.9e8
#Mass speed (at x=0 and t=0)

m0=1.
#Rest mass

k=1.
#Spring constant

PeriodNonrel=2.*math.pi*math.sqrt(m0/k)
#Non-relativistic period

dt=PeriodNonrel/n
#Time between computation loops

x=0.
#Initial position of mass

gamma=1/math.sqrt(1-v**2/c**2)

f=gamma*m0*g-k*x
#Total force on mass

a=f/(gamma**3*m0)
#Acceleration of mass

t=0.
#Initial time, when x=0

while
v>=0.: #Loop to
determine capture or escape

t=t+dt

x=x+v*dt+.5*a*dt**2

v=v+a*dt

gamma=1/math.sqrt(1.-v**2/c**2)

f=gamma*m0*g-k*x

a=f/(gamma**3*m0)

if(v<0):

print('mass is captured')

elif(v>.99999999999*c):

print('mass escapes')

break

print('x=',x,'v/c=',v/c)

print('End
of program')