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,
Because its "per", and so you are dividing. which gives you a negative
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 m
So 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.
"That which can be asserted without evidence, can be dismissed without
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.
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.
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.
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
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.
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.
s = ut + 1/2 at^2
m = m.s^-1 . s [ie m] + m.s^-2 . s^2 [ie m]
Those are good examples.
Dimensional analysis used to be included in Physics, I’m not sure if it was
at O or A level, it is sometimes difficult to recall when you learned
something you seem to have ‘always’ known. Speaking to a colleague when I
was still teaching, it was included superficially in Physics (I think A
level but possibly GCSE) but he was delighted when he learned I taught it
in my classes.
It goes hand-in-hand with checking that the answer you get is *roughly* what
you'd expect. If you calculate the volume of a hot-water cylinder and you
get an answer of a couple of millilitres (a thimbleful) or a million litres
(a swimming pool) then you've probably made a mistake ;-)
I find just being able to think in terms of dimensions - even if not
doing an accurate analysis on paper, it a very useful tool to catch
errors, and would also stop lots of silly mistakes.
Like how many times you see people talk about kW when they mean kWh or
vice versa. Just a basic understanding that the unit of energy is the
Joule, and that power in Watts is a measure of the rate of flow of
energy. Hence 1 Watt equates to one J/sec. Or energy over time. If you
then multiply by time, the two times dimensions cancel out and you are
left with just energy.
On Thursday, 12 September 2019 17:26:28 UTC+1, John Rumm wrote:
but for what they are talking about it rarely matters to average users.
What does the Kwh mean on fride/freezer consumption.
such as 124Kwh
note that oa.com uses an uppercase K
argos use kWh/year lower case K updercase W
When buying a kettle I found the figure of 3kw more useful than the fridge version and more useful than 10.8 mega joules of energy.
calories is an even more weird one.
a drink I have here per 100ml is 6kJ or 1 Kcal
an average adult 8400 kJ or 2000 kcal
Is that 2000 calories or 2000 kilo calories which is 2 mega calories :-)
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
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"
Yes I understand what it means: m.s^-2 or kg.m^-3 are identical to m/s^2 or
kg/m^3. All I was saying is that the former is a slightly cack-handed way of
expressing it (IMHO) and one which has to be translated back again if you
are going to refer to it in spoken form. But it does make it easier to check
that powers of units match throughout an equation.
its to do with the fact that mathematically speaking:
1 / x is equivalent to x^-1
1/x^2 is equivalent to x^-2.
so once the 1 is replaced by another quantity we have:
for density say:
kg / m^3 is equivalent to kgm^-3
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