Somebody with more skill at rocket-science than me has provided an even more ferocious condemnation of Richard Hoagland's faulty mathematics in attempting to calculate the velocity addition of Stages 2,3 and 4 of Explorer 1.
He points out that, using Hoagland's notation, Konstantin Tsiolkovsky's rocket equation is normally written as:
dV = g x Isp x ln(Wi/Wf)
where Wf is the final mass at burnout (but before discarding the burnt-out stage)
He then points out that it's not permissible to aggregate the stages and evaluate the equation once. You are obliged to take it stage by stage, and this is his result:
Wi = 1020+280+80
Wf = 1380 - 530 (weight of the burned fuel)
dV = 32.2 x 220 x ln(1380/850)
= 7084 x 0.482
= 3,414 ft/sec
Wi = 280+80
Wf = 360 - 140 (weight of the burned fuel)
dV = 32.2 x 235 x ln(360/220)
= 7567 x 0.492
= 3,723 ft/sec
Wi = 80
Wf = 80 - 48.5 (weight of the burned fuel)
dV = 32.2 x 235 x ln(80/31.5)
= 7567 x 0.932
= 7,052 ft/sec
3414 + 3723 + 7052 = 14,189 ft/sec
An additional 600 ft/sec represents 4.2% over-performance.
There is no need to resort to an anti-gravity field to account for this. Hoagland's entire thesis is therefore falsified. Cheers.
There's another, possibly simpler, way of looking at this. Since the parameters of the elliptical orbit are precisely known, the orbit velocity can be calculated from a reliable equation. Deduct the burnout velocity of the first stage -- also known, but with less precision -- and what's left is the dV of the solid upper stages. Calculations done this way were offered to the "Dark Mission" blog, but blocked. Hoagland and Bara accept nothing that challenges their ridiculous and misguided ideas.
Richard Hoagland writes that the solid upper stages of the Juno 1 rocket that launched Explorer 1 only contributed 3,520 ft/sec of the total orbital injection velocity. Well, let's see now...
Planned orbit 220 x 1,000 miles (352 x 1,600 kM)
Actual orbit 223 x 1,592 miles (357 x 2,547 kM)
Radius of Earth 6,375 kM
Gravitational constant, µ, of Earth 398,660 kM3/s2
Cut-off velocity of Jupiter-C rocket (Juno's liquid first stage) 9,020 mph = 13,229 ft/sec 
semi-major axis of actual orbit, Lsmaj, (357+6375+6375+2547)/2 = 7827 kM
distance from center of Earth to orbit point, R, 6375+357 = 6732 kM
velocity at orbit injection, Vorb = √(µ(2/R - 1/Lsmaj)) 
2/R - 1/Lsmaj = 0.000169
Vorb = √67.493 = 8.215 kM/sec = 27,111 ft/sec (actually only 2.5% greater than planned)
less the velocity achieved by the first stage: 13,229 ft/sec
Ladies and gentlemen, get out your calculators! In about 3 seonds you will falsify Hoagland's entire theory.
There's a small discrepancy between the two calculations, to be sure -- due to uncertainty about the velocity at first-stage burnout, perhaps.
Other factors that would be considered in a really accurate calculation:
* Correction to the value of g as the rocket ascends away from Earth -- a small positive vertical increment
* Boost given by rotation of Earth -- separately calculated as 1100 ft/sec horizontal
* Gravity drag -- perhaps negative 1000 ft/sec vertical
However, the discrepancy is nothing to compare with the difference between 14,000 and 3,520.