So, based on the video, your descriptions, and the comments/suggestions, you have a formula to prove and (if you are interested), a new implementation to try.
If the video depicts your system and you are looking only to prove the formula, then let's look at a few things.
If we are under the assumption that the second half of the video represents the system you are working on/installed, then you have a three part move:
1. The bridge starts in the middle and begins a forward move at a speed equal to the conveyor speed until it reaches its front/tunnel discharge EOT (basically, a length of BTD/2). EOT means end of travel, or a stop position, if you will.
2. It then switches to the calculated speed you mentioned in the first post and moves reverse until it reaches its rear/tunnel entrance EOT (a length of BTD).
3. Finally, it switches back to a forward move at a speed equal to the conveyor speed until it returns to the middle position (a length of BTD/2).
A simple analysis of the formulas shows that your physics buddy isn't considering the units, so there would need to be more to their equation to be correct. The original formula results in a value of X inches/sec. The suggested formula results in a value of Y (inches/sec)/(W inches - Z inches/sec), so without some time component for W (the car length - 28inches), you are stuck. The original formula did have an overshoot, but it is evidently a design feature.
Consider this example:
Assume: Conveyor speed = 600 in/min, Car Length = 1072 in, Bridge Travel Distance (BTD) = 100 in
1. Numerator: Conveyor Speed (in/sec) x BTD (in) -> ((600in/min)/(60sec/in)) x 100in = 1000in2/sec [1000 inches squared per second]
2. Denominator: Car Length + 28in - BTD -> 1072in + 28in - 100in = 1000in
3. Result: (1000in2/sec)/(1000in) = 1 in/sec
4. Conclusion: the bridge would need to move at a speed of 1 inch/second to reach its end of travel roughly at the rear end of the vehicle. I say roughly because in my example, the bridge will take 100 seconds to complete its reverse move of 100 inches and the car will have moved 1000 inches during that time. Relative to each other, the bridge moved 100 inches against the motion of the car, resulting in a relative difference of 1100 inches, so the 72 inches of remaining car length were actually surpassed by 28 inches (your PE length).
That 28 inches is actually shown in system in the second half of the video. It is a design feature for ensuring that when the third stage of the move is executed the spray nozzles get the back of the vehicle. As the bridge moves along with the vehicle during that third stage, it's really just spraying the same area of the car. When it finally stops in the middle position, the nozzles continue to spray, getting the rest of trunk/rear.
So, my final verdict is that your original formula works. The other one is incomplete. The 28 inches and 5 inches are factors you can add/remove/tweak to maximize the performance of your unit.
The only question I can think of is why it didn't work "in the shop," as you said. What was wrong with it there?