[FlamingCobra]

Here is our hypothetical situation: We have a cube in three dimensional space. On this cube, the acceleration of gravity is 9.8 m/sec².

If we bump this cube up to a hypercube in some way (perhaps using a hyperdrive), would the acceleration of gravity become 9.8 m/sec³?

[Scotty]

That's not how it works.  Gravity is a conservative force that acts on the object regardless of what dimension the object is.  For an object at approximately sea level on Earth, the acceleration due to gravity is approximately 9.81 meters per second per second.  That's a unit of acceleration, which is the rate of change of the velocity of the object, which is in turn the rate of change of the position of the object.  It is in no way related to what the actual object is.  Neglecting air resistance, of course, which doesn't actually affect the force acting on the object due to gravity, merely acts in the opposite direction to slow the accleration.

[General Battuta]

Here is our hypothetical situation: We have a cube in three dimensional space. On this cube, the acceleration of gravity is 9.8 m/sec².

If we bump this cube up to a hypercube in some way (perhaps using a hyperdrive), would the acceleration of gravity become 9.8 m/sec³?

No, you're misunderstanding what the units mean. Also you don't seem to know what a hypercube is...?

Scotty gave a good explanation.

[FlamingCobra]

A hypercube is a four dimensional analogue of a cube. It's like taking a cube and folding it in on itself at right angles..... in a way. We don't see things or think of things in four dimensions so it's difficult for us to visualize it. It's also called a tesserect. I think. Some animations I have seen appear to show the inside phasing in and out of the cube but it's supposed to actually be rotating. Don't know how that works.

No, you're misunderstanding what the units mean.

That's possible. But we were using derivatives in word problems that involved position, velocity, and acceleration. And we often use the "power rule" to get derivatives. And when you start switching around exponents and then your ADHD brain starts thinking about hypercubes....... that's what happens.

Also, thank you guys for taking the time to explain these things to me.

[Dragon]

This could perhaps work like that if you added a second time dimension, though I'm not sure, I'm only proficient enough in multidimensional calculations to be able to roughly imagine a 4D object without going insane. 
In general, when you bump the amount of "normal" dimensions, time stays the same. Gravity depends on mass, so assuming that mass of the cube and hypercube are the same, so will be their acceleration of gravity. Geometry of the hypercube's field might, of course, be completely insane (even for a normal cube, it's a bit awkward and hard to calculate).

[Scotty]

A hypercube is a four dimensional analogue of a cube. It's like taking a cube and folding it in on itself at right angles..... in a way. We don't see things or think of things in four dimensions so it's difficult for us to visualize it. It's also called a tesserect. I think. Some animations I have seen appear to show the inside phasing in and out of the cube but it's supposed to actually be rotating. Don't know how that works.

No, you're misunderstanding what the units mean.

That's possible. But we were using derivatives in word problems that involved position, velocity, and acceleration. And we often use the "power rule" to get derivatives. And when you start switching around exponents and then your ADHD brain starts thinking about hypercubes....... that's what happens.

Also, thank you guys for taking the time to explain these things to me.

Acceleration due to gravity doesn't care what dimension you are.  A flat piece of paper (2-dimensional object, for all intents and purposes) will still be acted upon with exactly the same force due to gravity as a hypothetical hypercube in the exact same location.  The only difference would then be the force in the opposite direction due to air resistance, which has to do with a lot of things related to physical properties of the object, but doesn't not actually affect the force due to gravity in any way, shape, or form.

[watsisname]

I think you're going along the line of thought that since the derivative of a position function is velocity, and the derivative of velocity is acceleration, that therefore the derivative of acceleration is something else.  And indeed it is -- it's the change of acceleration with respect to time.

This doesn't have anything to do with 4-dimensional space or tesseracts, though.  If we're describing motion in 3-space, then the position, velocity, and acceleration all use a 3-dimensional coordinate system.  There's no need to invoke a fourth dimension to describe da/dt.


If you want to start thinking about higher dimension, wait until you use integration to compute areas and volumes.  The integral over a curve gives you the area under the curve, the double integral over a surface gives you the volume under the surface, and you can extend this concept to higher dimensions and the triple (or higher!) integrals to find hypervolumes.

[Scotty]

Put slightly more simply: the acceleration due to gravity at a certain point on Earth is constant for every possible object that could occupy that certain point.  The units that you came up with earlier, meters per second per second per second, is the rate of change of the acceleration, referred to in physics as the "jerk" of an object.  While it is true that as you get closer to or farther away from the center of the Earth, the acceleration due to gravity increases or decreases, respectively, the effect is so miniscule that unless you're dealing with changes in position of thousands of meters per second straight toward the Earth (and then from a fairly low altitude, which means into the Earth, and practically impossible for our purposes), the jerk of the accleration due to gravity is negligible.

[watsisname]

A hypercube is a four dimensional analogue of a cube. It's like taking a cube and folding it in on itself at right angles..... in a way. We don't see things or think of things in four dimensions so it's difficult for us to visualize it. It's also called a tesserect. I think. Some animations I have seen appear to show the inside phasing in and out of the cube but it's supposed to actually be rotating. Don't know how that works.

Ahhh, that strange behavior arises because you're not actually "seeing" the tesseract, but instead merely a 3D projection of it.  As an analogy, observe the 2D shadow of a wireframe cube as you rotate it -- you'll observe similarly "strange" behavior because you're only seeing a cross-section of the full object.

Try reading Flatland, it's a great short story and may help you gain a little more insight when trying to understand higher-dimensional objects and spaces.
Also if you just want to mess around with visualizing 4D stuff, a nice program is Jenn3D.

[TwentyPercentCooler]

Also, I'd just like to add that we already have definitions for what exactly a m/s³ would be. Change in acceleration over time. It's usually referred to as jerk. We even have a term for the fourth derivative of position, or (in SI units) m/s⁴. It's generally known as jounce, or the change in jerk over time.

These values are sometimes considered in aerospace engineering applications.