Help with Physics signs and directions

love2learn1

The acceleration due to gravity is -9.8m.s^2 [U], so what does the up mean, and what does the negative mean? I thought that the negative sign always means down and you don't need to put the direction in brackets.

Simon Bridge

The author wrote [U] to tell you that "up" is the positive direction.
This means that -9.8m.s^-2 could mean the object is going down and gaining speed or it is going up and losing speed (or momentarily stationary).

The sign convention is arbitrary - you don't have to call "up" positive and in many situations it makes the math easier to make down positive.
You don't have to make the x axis horizontal either, or the y axis vertical.

But you do have to say somewhere what convention you are using, and you have to be consistent within the problem.

love2learn1

But if it says - and [up] doesn't that mean that up is the negative direction

Doc Al

That [U] seems an odd bit of notation. Where did you see it?

Since you know that the acceleration due to gravity is always down, if it's given as negative that means you are taking up as positive.

flatmaster

The notation does seem odd. He's using [U] where a unit vector would be used. Unit vectors have multiple conventions, but I've never seen [U]

x hat, y hat, z hat
i, j, k,
e1, e2, e3

nasu

They use similar notations in high school physics. In US and Canada.
[E], [N], [W], [S].

BruceW

What the author wrote makes sense to me. That is how I would say it, if I were explaining it with words. "In SI units, g equals minus nine point eight in the 'up' direction". And then if I define 'down' to be the negative of 'up', then it is equivalent to saying: "In SI units, g equals nine point eight in the 'down' direction"

Doc Al

Yep, it looks like they are using [U] to represent a unit vector in the up direction, like flatmaster pointed out. Makes sense to me, though I don't recall seeing that.

love2learn1

So does this mean that up was not chosen as the positive direction? I'm really confused on the sign conventions in physics. Why do you need the [U]?

jtbell

It means that "up" is the positive direction. Therefore "down" is the negative direction, and since the number given has a - sign, it indicates a downward acceleration.

If they had written it as "9.8 m/s^2 [D]" it would have meant the same thing.

I've never seen this particular notation either, but it's been more than forty years since I was in high school. The textbooks I've used and taught from at college level would say something like

$$\vec a = -9.8 \hat y \text{ m/s^2}$$

where ##\hat y## is a unit vector that points upwards. Some books use ##\hat x##, ##\hat y## and ##\hat z## for the unit vectors, others use ##\hat i##, ##\hat j## and ##\hat k##. Or they use boldface instead of the caret ("hat").

Simon Bridge

Beginning students often get confused when a negative sign is used with a positive direction.
The sign convention is the same in physics as it is in mathematics.
It indicates a deficit of something.

-10m [U] is a deficit of 10m in the upwards direction ... which is the same as saying "10m downwards".

You could just say, "well why not just say so then?"
The reason is because it makes the math easier when you refer to only one positive direction.
If you went 10m [U] and then 20m [D] you'd end up 10m [D] ... which you did intuitively.
In math you'd have to convert one of the [U] or [D] first so you can say (taking [D] as positive)
(-10m[D]) + 20m[D] = (20-10)m[D] = 10m[D]
In math, 10m[D] is "ten meters multiplied by the direction [downwards]".

Unless it says something else in the book (or wherever you saw the notation) then .. no. Treat the indicator in brackets as the positive direction.
In the case of [U] - that would indicate that "upwards" is the positive direction.

A negative value of a positive direction is the same as the positive value on the negative direction.

Using math notation we can say "down is the opposite of up" like this:

[D] = -[U] so:

-9.9[U] = -1x9.8x[U] =9.8x(-1)x[U] = 9.8(-[U]) = 9.8[D]

Because the direction of the positive axis is entirely arbitrary - if a direction is not specified, then the reader could get confused about what was intended. Maybe it seems obvious to you know, which direction things like gravity should be? Why bother saying it? Isn't it just being pedantic?

But if the problem is, say, a box sliding down a slope, then it makes the math easier to put the +x axis pointing down along the slope. This means the -y axis does not point in the direction of gravity.

If the author did not tell the reader that the +x axis was taken to be "down along the slope" then the reader may not realize that and get confused.

Somewhere where the reader can see it there has to be some indicator.
You could write: the initial velocity is -23m/s [down along the slope] with acceleration +9.8/2m/s/s [down along the slope] or just say +x dir = [down along the slope] (I'd use the last one personally.)

Whatever you use - it has to be clear to the reader, which is why I suspect the author of the notation you are referring to has actually explained it someplace and maybe you just missed it? Whatever - it is a failure to communicate. Learn from it.

sophiecentaur

It's all a matter of context. Anyone starting on any calculation of this sort is free to use either convention they want - as long as they stick to it throughout.
If we do calculations about falling bodies (down mineshafts and off cliffs) we often choose 'downward motion' to be positive and, hence the acceleration due to gravity will be a positive value. We state (or strongly imply) that this is what we mean and then we stick with that convention and get the right answer (how far down, etc.).
If we do calculations on a ballistics problem, we would (instinctively) be pointing the gun in a direction we would call Positive (UP = positve this time) and then our initial velocity would be positive and the acceleration would be negative.