Answer;
There are two identities here;
Let the original auxiliary pressure be ‘Aux'.
Let the brake pipe reduction pressure be ‘R'
Cylinder pressure be Cyl.
The amount of reduction in the brake pipe(R)= the amount of reduction
in the Aux.
Since the volume of air taken from the Aux is two and one half
times the volume of the
cylinder, and if the full amount of air taken out of the Aux,
(R), is squeezed into the cylinder,
then; at any brake pipe reduction, R, Cyl= 2and ½ xR.or
5/2xR
No more air will flow from the auxiliary to the cylinder once
the Cyl=Aux-R,
this is called a full set brake.
So for a full set brake,
Cyl = Aux-R
Substituting,
5/2 x R = Aux - R
Isolate Aux by adding R to each side;
(5/2R) + R = Aux
(5/2R) + 2/2R = Aux
or
7/2 x R = Aux
isolate R (multiply both sides by 2/7) then;
R = 2/7 x Aux ( for a full set brake )
If Aux on the head end = 84psi, R= 2/7 x 84 = 24 psi and since
all is equalized,
Aux - R = Cyl = 84 - 24 = 60psi in the cylinders on the head end.
On the tail end, Aux = 77. And 2/7 x 77 = 22 psi reduction for
equalization.
Aux - R = 77 - 22 = 55psi in the cylinders on the tail end.
The ratio of deceleration, discounting all else, is 55 to 60 (tail
end to head end ).
So during the time the head end decelerates from 60mph to 40 mph,
a deceleration of 20mph, the tail end has decelerated to 60mph
- (20 x 55/60)
= 60 - 18 and 1/3 = 41 and 2/3 mph.
When the head end reaches 20mph (deceleration is 40) , the tail
end is going
60 - (40 x 55/60) = 60 - 36 and 2/3 = 23 and 1/3 mph,
( 3 and 1/3 mph faster than the head end )
Now if we add to that difference, the time involved in response
from the head end
to the tail end, we have an even larger buff force! If you were
on a curve, at the time,
Look Out !! That is a more important reason for limiting your
speed on curves than
just the usual forces we come to think about.
Note: These are formulae that I scribbled out while
sitting at a lot of red signals,
They are not sanctioned by anyone but me, so please feel free
to correct or add to any of this.