Hello,
When writing, for example, "metres per second per second" for acceleration, it is noted as ms^-2 (where the caret symbolises the -2 is in superscript).
I understand that, and why, "per second per second" is "seconds squared", but in the notation, why is it superscript minus 2? Why not just superscript 2?
If I was ever taught this in maths lessons, the info is no longer in my head.
Thanks in advance for any help,
David Paste.
Because its "per", and so you are dividing. which gives you a negative coefficient.
E.g.:
Miles/hour is speed, and if I say I go at a certain miles/hour for a certain time, the answer must be miles. So, 30 miles/hour for two hours is 60 miles. That is:
30 m/h x 2 h = 60 mSo the /h and the h cancel, meaning that as the h is really h^1, the /h must be h^-1 in order for the coefficients to add up to zero (meaning there's no h left in the final expression). When you multiply things you add their coefficients, as in 10^2 x 10^3 -> 10^5.
because it is *per* second - the inverse, ie
10m--------------- second * second
1/x == x^-1 in this notation.Compare, for instance, measuring carpet, which is in meters^2, ie 'square meters', meters * meters, and not a minus in sight.
HTH J^n
'per second per second' is not 'seconds squared', it's 'per second squared' (perhaps you meant that but omitted the 'per').
'one tenth' or 1/10 can be written 10^-1. Metres per second per second can be written m/sec/sec or m/sec^2 or m.s^-2, the -ve sign indicating 'per' or 'one over' or division.
It's because it's /per/ second, indicating dividing. ms^2 would be metre- second-seconds (i.e. distance times time times time), while acceleration is ms^-2, metres per second per second, distance divided by time divided by time. Positive powers are multiplication, negative powers are division (or "anti-multiplication"). x^2 is x*x, x^-2 is (1/x)/x. 2^2 is 4, 2^-2 is
0.25.If you multiply acceleration (ms^-2) by time (s, or s^1), you add the powers - and get speed in ms^(-2+1) or ms^-1. If you divide acceleration by time, you subtract the powers - and get jerk in ms^(-2-1) or ms^-3. The same applies starting with distance, in m, or ms^0. Anything to the power zero is 1, so m and ms^0 are the same thing. Divide by time and you get ms^(0-1), ms^-1.
Mike
Its a bit like the way you can do a division by multiplying by the reciprocal of the devisor...
So m/s^2, can become ms^-2, so its including the "per" (i.e. the division operation) into the exponent.
because 'per'
meter seconds squared is not the same as meters per second squared
The notation m s^-2 and kg m^-3 always strike me as perverse: what was wrong with m/s^2 and kg/m^3? I remember being taught m/s^2, and then when I changed school to one that did a Nuffield physics syllabus, the notation changed to kg m^-2 which it claimed to be "better" in some unspecified way.
I'm inclined to agree.
While all (or perhaps most) of the answers posted are correct, I don?t think any quite get to the basic reason.
For that you need to look at the laws of indices.
4 = 2^2 2= 2^1 1 = 2^0 1/2 = 2 ^ -1 1/4 = 2^ -2 = 1/(2^2)I?ve use 2 for simplicity but the rules apply for other numbers.
Hopefully you can see the pattern.
If we substitute m for 2 then, in particular in the last line we get:
m^-2 = 1/(m^2) which is also 1/m * 1/m
The laws of indices ?pop up? in a number of places and can be very useful. They are the basis of Logarithms, can be used to find HCF and LCMs, .......
I disagree. Mainly as the superscripts reduce the risk of mistakes where there are multiple terms. Eg
kg.m^?1.s^?2 kg/m.s^2
Certainly if you are analysing a problem and checking the dimensions ( a useful technique, not always taught these days although I made a point of teaching it), indice notation is probably far easier to use.
I suspect it is a question of what you are used to. A physics teacher when I was a pupil first introduced indice notation for units, so it is the one I?ve used most. I have used others but much prefer indices.
My understanding is that this is why the "simpler" notation is deprecated. Also it is better when you are typing stuff from scratch. In the old old days where manuscripts were hand-written and subsequently transcribed by typists, or you were exchanging hand written calculation sheets with colleagues the slash notation is slightly easier to read. Especially if you express the formula as a fraction, with a full horizontal line.
Yes.
I never liked subtraction (or division) because it was non-commutative as opposed to addition (or multiplication) which are commutative. Commutative operations are far easier to handle algebraically.
When my son was little I taught him addition of negative numbers rather than subtraction, he liked the idea. You were a teacher, weren't you? I always wondered why this approach wasn't more common in schools?
Indicies emphasise multiplication by an inverse instead of division and hence are commutative. Personally, I don't seem to care about the notation, I use both.
As long as you imply brackets according to the standard BODMAS rules - ie kg/(m.s^2) rather than (kg/m).s^2 - then you'll be OK in either notation. So in slash notation you put all the units that are multiplied before the slash and all the units which are divided (ie have a negative index) after the slash. Accelerations in m/s/s looks absurd but if you parse it as (m/s)/s then it's obvious that it's a rate of change of speed wrt time; mind you, even I wouldn't go so far as to quote accelerations that way.
I think the slash notation tends to match more closely how you would say the units - "metres per second" rather than "metres seconds-to-the-minus-one" ;-) Index notation is for purists (like IUPAC names for organic compounds) whereas slash notation is for everyday use (like pre-IUPAC names): most people know that vinegar contains acetic acid, whereas only people who have studied chemistry will know that its "Sunday School name" is ethanoic acid. The latter is logical and makes it obvious that it contains an ethyl radical like ethane and ethanol do, but it's not the common "lay public" name.
At least no-one's perpetrated the ultimate howler: whereas it makes sense to refer to a volume (kg/m^3 or kg m^-3) in "kilogrammes per cubic metre", it is ludicrous to refer to an acceleration in "metres per square second". The former uses a unit of physical linear dimension, and everyone can visualise an area in square metres or a volume in cubic metres, but "square seconds" is ludicrous.
There are several techniques.
Opinions on which are easy to grasp vary.
The same is true for other areas of Mathematics. This is why, for example, long division wasn?t taught for many years in primary schools which has had a ?knock on? impact even now. Some primary teachers can?t handle long division with confidence so avoid it (they never learned it themselves).
You may think, so what but it is key to algebraic division which you need later. Yes, I was a teacher, in a Specialist Mathematics School. I qualified late in life as a second career, I?ve degrees in Engineering and another in Mathematics. I always tried to bring real world examples into my lessons.
Once you are confident, coping with different notation etc isn?t generally an issue. Ditto terminology. However, when people are learning something or less confident, consistency is important.
I agree about the usefulness of dimensional analysis.
If I can't quite remember a formula, I use dimensional analysis. Is the dimension that the formula produces different to the dimension of the answer that I would expect? Am I adding together terms which are not the same dimension? If so, check that I've remembered the formula correctly.
v^2 = u^2 + 2 as
m^2.s^-2 = m^2.s^-2 + m.s^-2 . m [ie m^s.s^-2]
So if I had brainfade and mis-remembered it as v^2 = u + 2as, I'd soon realise that something was wrong.
Likewise with
s = ut + 1/2 at^2
m = m.s^-1 . s [ie m] + m.s^-2 . s^2 [ie m]
it's metres ;-)
It normalises notation to exponential form, that is all. The - in the exponent is equivalent to the / of division
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