If an object, a human for instance, were to slide down a never-ending frictionless slope, would they eventually accelerate to terminal velocity? Or would the fact that they are not traveling straight down or the degree of slope affect their maximum velocity?
The fact they aren’t gong straight down wouldn’t matter. If there’s also no air resistance, there wouldn’t be a terminal velocity.
As you go lower in a gravity well, your potential energy has to go somewhere. If there’s no friction, it can only go into kinetic energy, hence continuing to accelerate.
The acceleration on the slope would be less than falling straight down. Eventually however the speed of light would be reached, and acceleration would stop.
“Terminal velocity” means the speed where the retarding force due to air resistance matches the accelerating force due to gravity. Once free-falling up to that speed, you will not continue to accelerate. (Ignoring issue like getting deeper in the gravity well so gravity is increasing, and issues like getting deeper into the atmosphere so air resistance is increasing.) The concept of terminal velocity also assume you started falling from stationary. It doesn’t apply to things like re-entering spacecraft or meteors.
So the question turns entirely on what sort of atmosphere & gravity you assume are around your frictionless slope. And on what we’re dropping/sliding down the slope. Feathers and lead balls have different terminal velocities. People have different terminal velocities depending on their posture while falling. (And their obesity.)
Enough back story & caveats, on to the solution …
The slope-wise acceleration due to gravity is proportional to the sine of the slope, where 0 degrees = horizontal & 90 degrees = vertical. So on a shallow slope, say 5 degrees, the acceleration would be only roughly 8% of the typical vertical acceleration.
So on any slope other than 0 = horizontal, you’d still reach a “terminal velocity” where air drag was enough to counteract acceleration down the slope due to gravity, but for our 5 degree example it would be much slower than of the usual vertical “terminal velocity”. It’d be ~5-10% of normal.
On a 30 degree slope acceleration would be half of typical, and on a 45 degree slope it’d be ~70% of typical. Because air drag is proprtional to the square of the speed (all else equal), 70% of typical acceleration will not produce 70% of the typical speed. It’ll be more like (WAG) 90%.
Why would it not apply to these? An object that starts out above terminal velocity experiences deceleration until it reaches T.V. (or, perhaps, burns up).
Just to confuse things there is another definition of terminal velocity. T.V. can also be the opposite of escape velocity. That is there is a limit to how fast a gravity can accelerate an object. Since our OP assumed an infinite frictionless slope, I assumed an infinite constant source of gravity. Why not? Its just a though experiment. Also I took frictionless to mean no friction at all, so I ignored wind resistance.
To say it simply, the answer to this question depends on what assumptions you make.
More pedantry: assuming the gravity source (attracting mass) is at the other end of the infinite slope, you won’t slide down, since the force of gravity will be 0. You also won’t be able to push yourself towards the source, since the slope is frictionless.
No, you can never reach the speed of light. You’d continue accelerating for ever. To you, you’re be accelerating at the same rate you were from the beginning. To an outside observer, your rate of acceleration would have decreased so that you never actually reach the speed of light.
I think that is what figure9 means.
Thank you for pointing out my error in such a gentle fashion. My understanding of relativity is minimal, and I always get confused on frame reference. I just can’t get my mind around relativity. I love science and read up on it, but I am a rank amateur at best.
Or perhaps impacts the surface before slowing significantly.
What I really meant was to short circuit a large discussion of those loose ends with as short a sentence as possible. …
Several posters over the years have interpretted “terminal velocity” as some sort of speed limit in the atmosphere. Since our OP was obviously not much good with Physics 101, I wanted to cut that particualr rabbit trail off before somebody launched a side trip into the dynamics when starting from other than the rest state.