To comment more on my post 74 above, I dug out my Kleppner and Kolenkow mechanics textbook from college, and the path neglecting the earth in the rest frame of the moon is an unbound hyperbolic trajectory. Given the speed at the minimum point was 3140 m.ph. at a distance of 5170 miles from the center of the moon, the entire trajectory can be worked out in the reference frame of the moon. Knowing the moon's approximate speed in orbit around the earth, I used 2250 m.p.h., one can transfer back to the earth's reference frame, and what is found, also using energy considerations in the frame of reference of the moon (in its rest frame), that the initial velocity when the rocket crossed the path of the moon was almost straight ahead at somewhere between 950 and 1000 m.p.h., which in the rest frame of the moon becomes at an angle with a speed of about 2650 m.p.h. In the rest frame of the moon, this speed increased to 3140 m.p.h. from left to right, but going back to the earth's frame of reference, it is just under 900 m.p.h. It appears the earth slowed the moon by about 20 m.p.h. per hour and the calculations support this.
I have some Nasa data of the flyby from about 4:00 PM to 5:00 PM CST that Wednesday, but I'd be interested in getting a data point at the time when the rocket crossed the moon's path, which I estimate to be around noon. The distance could be interesting. My calculations show, (using the hyperbola), it should be just under 17,000 miles from the center of the moon, or just under 16,000 from the moon's surface. I also think the rocket may have had a slight velocity component from left to right at that point of perhaps 100 m.ph., but this may be beyond what the uncertainty in my calculations can accurately determine. I also might not have a perfectly precise number for the speed of the moon=something that the Nasa engineers no doubt knew very accurately. The earth also played a slight role in the trajectory, so it would no longer be a perfect hyperbola in the rest frame of the moon.
and note, contrary to what we might have guessed, the rocket did not pick up speed when measured in the rest frame of the earth in its encounter with the moon. Part of the reason is that the moon was off to the right from the rocket when the moon's gravity became stronger than earth's, instead of being directly in front of the rocket.
It may be worth mentioning, the hyperbola around the moon, which is the precise path without the effects of the earth, has a symmetry to it, so that the path after the point of closest approach is a mirror image of the path before it. In addition, the hyperbola has an asymptotic behavior to it, and for this case, when transformed back to the earth's rest frame, the path looks like a ribbon that crosses itself below the moon. The asymptotic speed in the rest frame of the moon is about 2000 m.p.h., with a y component of velocity of about 850, and an x component of about 1850, so that in the rest frame of the earth, it actually has an x component right to left of about 400 m.p.h. far from the moon, with this calculation being the result from the perfect hyperbola, using the maximum speed number at distance number at closest approach.
Once again, I could use the Nasa data from noon that day, if anyone has it.