BBC Focus Magazine

[Ush]

For example, the laws of classical mechanics (Newton's Laws) are valid only when the velocities of interest are much smaller than the speed of light (that is, in algebraic form, when v/c << 1).

Why in the above quote are two 'less than' ('<') symbols used?

It's very strange to me.

[The Beige Avenger]

means, much less than.

in this case the velocity must be significantly lower (much much less) than "c" in order for a non-relativistic approach to be made.

[Ush]

Thanks.

So by convention, how much less must one thing be to another before the '<<' symbol is used?

[Ush]

I have more problems…

What does this triangle mean: ∆x

I think it means error of x but need clarification.

[The Beige Avenger]

How much less? A few orders of magnitude probably... if you can subtract 'v' from 'c' and still have something close to 'c' then you can essentially ignore the velocity. You can of course include the velocity whenever but you'll end up with many numbers after the decimal point which you will round up / down which will pretty much cancel out your efforts of including the velocity at not relativistic speeds. (Try a couple of examples).

As for the triangle, that is 'delta' (upper case) and it means 'change'. Sometimes a lower case delta is used, sometimes a plane old 'd' and then you have a curly 'd' which means something similar only more specific (partial deriv).
it's used in calculus and physics heaps...
for instance: v=∆x/∆t, a=∆v/∆t
Are a more correct way of writing v=d/t, a=(v-u)/t

[jpYB3Gq]

Good reply from Avenger

[Ush]

Ok I get it. Cheers.

for instance: v=∆x/∆t, a=∆v/∆t
Are a more correct way of writing v=d/t, a=(v-u)/t

More correct or more cumbersome?

[The Beige Avenger]

Ok I get it. Cheers.
.........
More correct or more cumbersome?

It can seem they do it just to confuse us eh?

But...

Do you think "v-u" is less cumbersome than "∆v"?

It's to let us know we are interested in change, not absolutes. So, delta can be used to abbreviate 'value at start - value at end'.

[Ush]

I some times think they try to make easy things appear hard so as to frighten people off to protect their jobs or carve out a special niche for themselves.

[jpYB3Gq]

So, delta can be used to abbreviate 'value at start - value at end'.

Wrong. Delta is end-start.
Understanding has different levels, ush.

[Ush]

Wrong. Delta is end-start.

Pedant.

My post above this one was insincere by the way. I forgot the smiley.

[The Beige Avenger]

So, delta can be used to abbreviate 'value at start - value at end'.

Wrong. Delta is end-start.
Understanding has different levels, ush.

Jeez... got nothing to contribute?

It doesn't really matter anyway if it's end-start or start-end just as long as you are consistent.

[jpYB3Gq]

There is a big difference.
If on day 1, you have 100 pounds, and on day 2 you have 150 pounds,
end - start = income of 150-100 pounds = + 50 pounds.
start - end = income of 100-150 pounds = -50 pounds = a expense of 50 pounds.

In business, it is the difference between profit and loss.

[The Beige Avenger]

So...

day 1, 100
day 2, 150

start - end -> (100 - 150) / (day 1 - day 2) = -50/-1 = 50
end - start -> (150 - 100) / (day 2 - day 1) = 50/1 = 50

you ignored the passage of time.

Try harder.

[EzBloke]

Fight!

[The Beige Avenger]

There's no fight here although it is a little frustrating when it seems that there is deliberate belligerence and obstinateness to simplifications conveying a general point.

Change matters not if there is not something it changes relative to, in this case, relative to itself at an earlier point in time. Change can be relative to any common factor for the system... it need not be time, it could be distance, or whatever.

For instance, plot a graph, the data is nice and the straight line trend intersects all points. If we take the rate of change of the quantity on the y-axis (y2-y1) and divide it by the rate of change for the corresponding values (x2-x1) on the x-axis (dy/dx) we get the slope of the graph i.e. the gradient. Similarly if we take the equation of the line (y=mx+c) and differentiate it we get dy/dx = m.

If the order is reversed and you performed y1-y2/x1-x2 you get the same result.

[jpYB3Gq]

You have got it all wrong.
Velocity is the ratio of two deltas:
v = delta (x) / delta (t)
But delta (x) = new (x) - old (x) , delta(t)= new(t) - old(t)
Hence the general definition of delta is always new - old. This is very IMPORTANT!!!

I have got distinction for GCE O level Physics , Mathematics and Additional Mathematics , GCE A level Physics, Mathematics, and have a B.Sc Hon 2nd Upper Physics degree and I have been a school teacher for 1 year and have been privately tutoring students for more than 20 years. I am just concerned that you may be teaching the wrong thing.

[Shadowwolf]

I have got distinction for GCE O level Physics , Mathematics and Additional Mathematics , GCE A level Physics, Mathematics, and have a B.Sc Hon 2nd Upper Physics degree and I have been a school teacher for 1 year and have been privately tutoring students for more than 20 years.

Come now Mr jpYB3Gq, an argument from authority is hardly going to benefit your position is it.

[Esther]

Isn't it also possible that different countries use different notation?

I seem to remember having a few problems combining UK books, USA math tutoring software, and then Portuguese school.

Also for different applications it could be just that you need to know the 'size of the change' rather than also the direction of change?

[jpYB3Gq]

I am really concern about this!

A lot of students are going to end up doing their physics wrong, please spare them.
The crux of the issue is: the delta has a life of its own , independent of anything else.

Let me give you just two examples:

Eg 1: ( Mechanics)
If somebody has a delta(v) > 0 , it means that he is accelerating, and if delta(v) < 0 then he is decelerating. There is a high physical significance. I am talking from a very physics standpoint.

Eg 2: (Thermodynamics)
If a container of gas has a delta(U)>0 in a thermodynamic process, it means the gas internal energy increases, or if delta(T)<0 in a process, it means its temperature drops.

Students are going to face this type of thinking process, usually when they do exam papers such as multiple choice questions. This is from experience!

Of course delta(time) is always positive , otherwise you have time travel!