There are a couple of episodes of Star Trek that I can recall having some fundamental physics failures, which would lead one to believe that in the Star Trek universe, one cannot do an integral over time. The episodes that come to mind (and it’s been a while, so I may have some details wrong) are The Paradise Syndrome from ToS, and Déjà Q fom TNG. In both episodes, the Enterprise needs to transfer some energy and momentum to an object, and in each episode, they go for the Big Effort™ and lose.
In The Paradise Syndrome, Spock tries to deflect a large asteroid and fails to budge it, so he goes for broke and zaps it so hard he burns out a whole bunch of circuitry — the sci-fi equivalent of overexerting one’s self and pulling a muscle — and can subsequently only match the speed of the asteroid. It’s after this that we learn that the asteroid is two months away from the planet; a force exerted continuously for two months would transfer half a million times more momentum than their ten-second attempt, so they could have even tried a smaller force for that duration and deflected the asteroid. But that makes for boring TV. (And they could have increased their speed my throwing junk out of the rear shuttle bay, with bonus points if the projectile hit the asteroid, since the collision would slow it down. This would have been slightly more exciting than two months of pushing, but still not very much excitement) Similarly, the attempts to restore a moon’s orbit is made in fits and starts in Déjà Q, though in the plot there is at least an excuse for interruptions to their attempts, from some attacking Calamarains, but that’s after they gave up a few times. Forces cause accelerations and change momentum of an object. With the exception of the static frictional force on a surface, these don’t turn on and off only when a threshold is reached*.
\(p = int F dt\)
For a constant force this is just p = Ft. Linear in force, but also linear in time.
What’s the connection to politics? The US government seems to approach solutions to problems like the Star Trek folks do. Wait until the problem is a crisis and then try and exert a huge force to correct it, when a much gentler push would have sufficed if you had simply started earlier. We have been seeing this with Social Security for decades now — we know the system is going to go broke, and yet nothing is being done to fix it. Had we started when I first started paying into the system, the adjustments could have been relatively small. But like the transfer of momentum, the longer we wait, the force needed to achieve the desired result gets larger. The occasional nudge does only a little; it needs to be sustained.
Similarly for global warming. Our government hems and haws and does very little to actually address the problem. Even those politicians who are still doubtful (or whose palms are being greased so that they act doubtful) should be able to recognize that there is value in weaning our country from foreign energy sources, and that the kind of technology adoption involved takes decades to realize.
Of course, getting them to do something would be asking them to do their job, and we can’t have that, can we? Star Trek ignored physics because slow-and-steady makes for little drama, and TV, like sex, is all about having a climactic ending. Our elected officials have no such excuse. They are distracted by the manufactured controversy du jour, and are more concerned with not upsetting their benefactors and voter base than doing the business that’s in the best long-term interest of the country.
*which really isn’t how the frictional force behaves, but it’s a reasonable first-order approximation for its highly nonlinear behavior
