Is there a scalar acceleration?

Question

The magnitude of Velocity is simply Speed.

On the other hand, the magnitude of Displacement seems to be a simpler idea than Distance.

And the magnitude of Acceleration is not the change of Speed over time. Is there a scalar counterpart to Acceleration?

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Edit to Question

If I walk to the store and back home, my displacement will be zero, so, regardless of how long this took, my velocity will also be zero. Since the store is some distance away, my total distance traveled will be positive, so my speed will also be positive. Therefore, just as distance is some scalar different from the magnitude of displacement, speed is some scalar different from the magnitude of velocity.

It seems that distance is a summation of the magnitudes of a group of displacement vectors, while speed is a summation of the magnitudes of the time derivative of the group of displacement vectors. I doubt there is (or is there a need for) a similar summation of the magnitudes of the second time derivative of displacement vectors. I welcome your thoughts.

Answer 1

The way I read this question it's asking for an unambiguous word to use to describe only the scalar part of acceleration. To the best of my knowledge this word doesn't exist in the English language.

The thing is, Acceleration is a vector but the scalar part is also acceleration. The word is overloaded. Why? Well vector came along later.

vector (n.)
"quantity having magnitude and direction," 1846
vector | etymonline.com

acceleration (n.) "act or condition of going faster," 1530s
acceleration | etymonline.com

As you can see, in the English language at least, acceleration as a scalar without direction has a few years on the word vector. The problem is that we never introduced a word as the pair of acceleration to give an unambiguous way to distinguish between the two ideas. So when we started using vectors we just overloaded the word acceleration. You can use it to mean either one. Which means readers have to figure out the meaning from context.

If I walk to the store and back home, my displacement is zero

Well it is now.

and my velocity would also be zero (regardless of how long it took me). So long as right now you're holding still and not heading for the backdoor on your way to the pool.

Since the store was some distance away, my total distance traveled for this situation would be positive

True

and so my speed would be positive.

Er huh? You mean your average speed? Your speed now? These are not the same thing.

Distance is paired with Displacement and it seems to be a bigger idea than just the magnitude of Displacement. Speed is paired with Velocity.

I think this might be the root of the problem. This is not the same kind of pairing.

Rate of distance is speed. Rate of displacement isn't velocity. Velocity doesn't know or care where you started. Velocity doesn't know or care how fast you were going when you started. Velocity is about how fast you're going now and which way. No, rate of displacement is average velocity.

The difference between distance and displacement is that displacement is "as the crow flies". Displacement can be measured between two position points. Distance is measured over an infinite continuum of position points that represent everywhere you've been.

If I was going to organize these concepts they'd look like this:

--------------------------------------------------------------------------
| Measured at one point in time | Measured at two points in time         |
--------------------------------------------------------------------------
| Scalar        | Vector        | Scalar            | Vector             |
--------------------------------------------------------------------------
| Distance      | Position      | Displacement      | Displacement       |
| Speed         | Velocity      | Average Speed     | Average Velocity   |
| Acceleration  | Acceleration  | Ave. Acceleration | Ave. Acceleration  |
--------------------------------------------------------------------------

The difference between Speed and Average Speed:

It's possible to move 1 mile in an hour and at the moment that hour ends be going 60 miles per hour. This difference has nothing to do with dimensions. It has to do with driving like my Grandma.

For more about this look up the fundamental theorem of calculus

Answer 2

The magnitude of acceleration is scalar, same as the magnitude of velocity (speed) which is scalar. It's just that the magnitude of acceleration doesn't seem to be that useful a concept, so we don't have a word for it.

That's because speed tells you a lot about how quickly you overcome some distance in normal space and that idea is natural for us. Acceleration tells you the same thing, but in velocity-space instead of normal space and since we don't live in velocity space it is unnatural for us to think about it. But its the same concept.

Answer 3

g-force. Example: pilots worry about g-force, but not the direction.

Furthermore: in space-craft engineering $g$ is a commonly used unit. For instance, the Mars Exploration Rovers (the ones that were dropped on the surface in airbags) were designed to tolerate 40 g on the 1st bounce, and this (plus margin) was the mark to which internal components were tested. Likewise, sustained hypersonic entry and transient parachute-deploy decelerations were discussed in g.

Launch and thruster induced vibrations were quantified via acceleration spectral density in "g-squared per Hertz". I believe this also standard in earthquake-related structural engineering.

Answer 4

just as distance is some scalar different from the magnitude of displacement,

You mean that distance travelled does not equal the magnitude of displacement. This is true.

If I walk to the store and back home, my displacement will be zero, so my velocity will also be zero.

You mean that your average velocity will have been zero. This is true.

my total distance traveled will be positive, so my speed will also be positive.

You mean that your average speed will have been positive. This is true.

Therefore, speed is some scalar different from the magnitude of velocity.

You mean that average speed does not equal the magnitude of average velocity. This is true: $$\frac{\text{distance travelled}}{\text{time elapsed}}=\frac{\int^{t_2}_{t_1} |v|\:\mathrm dt}{t_2-t_1}\not\equiv\frac{\int^{t_2}_{t_1} v\:\mathrm dt}{t_2-t_1}=\frac{\text{displacement}}{\text{time elapsed}}.$$

Is there a scalar counterpart to Acceleration?

The rate of change of speed, which can be negative, doesn't equal the magnitude of acceleration, and neither has a special name. The former is the component of acceleration in the direction of velocity: $$\frac{\mathrm d}{\mathrm dt}\left|\vec v\right|=\frac{\mathrm d}{\mathrm dt}\sqrt{\dot x^2+\dot y^2+\dot z^2}=\frac{\dot x\ddot x+\dot y\ddot y+\dot z\ddot z}{\sqrt{\dot x^2+\dot y^2+\dot z^2}}=\frac{\vec v\cdot\vec a}{\left|\vec v\right|}=\vec a\cdot\hat{\vec v}.$$

Answer 5

Not a facetious answer, and not entirely general, but how about Gs (as in multiples of the Earth's surface gravitational acceleration)?