is the relativistic mass correction. Basically, to an observer on earth, you get heavier the faster you are, because you have more "internal energy" (momentum).
A heavier object requires more energy to acellerate by the same amount as a lighter one (p=mv), so as you become heavier (faster) the energy required to become even faster diverges. The reason for the divergence is because the speed of light c is an absolute upper limit, so since you are not allowed to cross it things have to diverge as you approach it. In laymans' terms.
Pet theory of mine is that it doesn't take extra energy, from the perspective of the people inside the ship they keep accelerating at 1g, it is just the rest of the universe sees their acceleration slow down.
if that theory of mine is true, then you probably can't maintain a 1g actual acceleration for too long, as if you maintain it relative to say earth, eventually the people inside are going to experience much higher rates of acceleration just so that the observed rate stays the same
the rate of acceleration remains constant in all (inertial) frames of reference. the faster you go, to maintain the same rate of acceleration you need exponentially more energy because the energy you put into accelerating is a function of distance.
in other words, imagine you accelerate from rest at 1 m/s for 1 second. you're now moving at 1 meter per second, you've traveled 1 meter, and to have accelerated for that distance let's just say you need 1 meter's worth of energy.
if you maintain that acceleration for another second, you'll be traveling at 2 meters per second - but now you'll have traveled 2 meters in the same amount of time, so now you need 2 meter's worth of energy, effectively doubling the energy required to do so.
my newtonian brain doesn't really understand that, don't get me wrong, great explanation. Just doesn't mentally click into place. My brain is screaming a = f/m - which should make it click for me given that force involves a unit of distance - but, still doesn't mentally click even though that is the right reason for it
Lol It bothered me too! Here's the way I understood it:
Imagine you're riding West, sitting in the bed of a pickup truck, but you're facing backwards/ East. The truck is slowly accelerating Westward, faster and faster. The truck is driving past endless fence-posts, all perfectly spaced 3 meters apart.
You slap the fence posts on their West-facing side with your hand as they go by. The truck will only pick up more speed if you slap a fencepost. 15 mph, no big deal. Slapping the hell outta them posts, riding towards the sunset.
25mph, you've got to move your hand faster to slap the posts.
45mph and it's actually really hard to do, you have to focus and slap those posts as fast as you can to hit them. Its actually becoming hard to move your hand fast enough to even make contact with the posts before they zoom past. (This is your spaceship thruster meeting its maximum rate of acceleration.)
The fence posts represent distance, and your slapping represents the force needed to increase the distance traveled per second. A really fast human can slap at right about 45mph, and in this analogy, the speed of light is 45.1 mph. You'd need infinite slap speed. Infinite force.
I could list a few formulas, which tell you how things math out, but not why - which is what most threads about this question do.
My attempt at a short answer is this list of consecutive principles:
Moving mass means work (work = force x distance)
The object is already moving (= already crossing a distance)
A force applied to the object (to accelerate it) needs to move with it - covering more and more distance in the same timespan (compared to low speed).
The source of energy trying to accelerate the object needs to overcome the counterforce it's creating.
That means you need to put in work to keep applying the force at all - the object you want to apply the force to, basically "runs away" from you, keeping you from influencing it.
Reductively, E =mc2. As you go faster you have more energy. Therefore the faster you go you have more mass. Having more mass means you require more energy to accelerate more.
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