...what the book wanted the student to do was show the work (formulas and calculations) that proved that the swimmer went nowhere.
Not that the swimmer went nowhere, but that the NET work that the swimmer did was zero.
Work has to do with mass being displaced and if there is no
net displacement, there is no
net work.
At any point along the swim, one could stop the swimmer and determine the amount of work done to that point. Or, one could determine the amount of work done by the swimmer between two points in the swim. But if the swimmer comes back to the original point then the
net displacement is zero and the
net work is zero.
https://www.britannica.com/science/work-physics
Imagine that you win a million dollars in the lottery and then lost it all gambling the same day. Your net worth didn't change when only comparing the beginning of the day and the end of the day. That doesn't mean you never won any money that day or never lost any money that day, it's simply an assessment of where you are at the end of the day vs. where you started the day.
Imagine that your job is moving a stack of bricks from one location to another. You move all the bricks, but then move them back to their original location before your boss sees them. Will your boss say you did a lot of work that day when you tell him you moved the bricks twice? Or will your boss say you did no work because the bricks are right where they started. Obviously you put forth a lot of effort, but the net effect is zero in terms of accomplishing your assigned work.
Physics is a fantastic thing, but there are places in physics where the math works but reality doesn't.
The value of physics is that it does provide insight into what happens in the real world. It is necessary to fully understand the concepts being used before it's possible to make sense of them, however.
A big problem with this is that the physics we learn in school is often simplified to make the problems tractable. So, for example, people often remember that all objects fall at the same speed, but forget that the problem was simplified by omitting air resistance. It is true that all objects fall at the same speed
in a vacuum, but it is absolutely not true that the all fall at the same speed in air.
SO, no matter what formulas you use, where you measure, and what you call it, since ALL the energy in the system comes from the firing of the round, how can recoil energy and muzzle energy NOT be exactly the same???
The simplest answer is that it just doesn't work like that.
Muzzle momentum and recoil momentum will be the same, but muzzle energy and recoil energy will not.
Unfortunately, at the moment, I can't think of a good way to explain why this is true in a way that would make intuitive sense.
You can calculate which load will, on paper, have less recoil in your gun. BUT, what you feel is what you feel, and its not impossible that what looks like a significant difference on paper might not be as big a felt difference, in your hands.
Felt recoil depends on a lot of things that have nothing to do with the simple physics of the gun's recoil. The best example of this I can think of is when I let a friend with small hands shoot a Beretta 92 pistol. She complained that the recoil was severe which I didn't understand until she showed me that it had burst a small blood vessel in her thumb causing bruising, swelling and pain. Because the grip was too large for her, it had recoiled against the knuckle of her thumb and broken a blood vessel. The grip fit my hand and so the felt recoil was negligible.
The recoil velocity and recoil energy were the same for both of us, but how it felt and how it affected the two of us was obviously very different. Not because the math or physics was wrong, but because there was more going on than just how fast the gun recoiled and how much recoil energy it had.