# Very simple question

1. Feb 25, 2012

### xzardaz

Hello , I have a very simple question :

Why do we mesure acceleration in $\frac{m}{s^{2}}$ ?
We all know that the speed is mesured in meters per second. This is very intuitive - it describes the speed as mevement of a point : how much distence ( in meters in the Si system ) the point travels per some amount of time ( seconds in Si ).

The acceleration is by definition the change of speed over time , so it is mesured in the mesurement unit of the speed per second.

So if we define a name for speed units , for example "louis" , such that 1 louis = 1 meter per second , we can mesure acceleration in louises per second. This is the same as ( meters per second ) per second.

I know , we can use math to simplify $\frac{\frac{m}{s}}{s}$ = $\frac{m}{s^{2}}$ and that's how it's done. But I wonder what gives us right to use math in this particular case , what is the phisical meaning of s2 ?

Why we don't mesure the acceleration in $\frac{\frac{m}{s}}{s}$ , instead - it have physical meaning for all variables ?

I know it is mathematicly the same , but I used to understand the sence of the physics mesure units , and now I don't see the logic of s2 ( It may have some meaning, but I don't see it ).

So why do we mesure acceleration in $\frac{m}{s^{2}}$ instead of $\frac{\frac{m}{s}}{s}$ ?

2. Feb 25, 2012

### tiny-tim

hello xzardaz!
because if we convert between seconds and minutes, we need to multiply by 602

that's really all we're interested in

3. Feb 25, 2012

### Michael C

If two expressions are mathematically identical, we know that all calculations will be identical when we replace one with the other. That's precisely why math is so useful. Of course we have the "right" to use math here: the validity of all scientific formulas depends on the basic assumption that "math works"!

There is no need for every term in an equation to have a "physical meaning". The formula for KE, for instance, is $\frac{1}{2}mv^{2}$. What does a squared velocity look like?

4. Feb 25, 2012

Suppose an object started moving from rest and accelerated uniformly for between one to two minutes.Now suppose at 7 seconds the speed was measured to be 70 m/s.By how much did the speed increase by?You couldn't give an exact answer because you don't know the total time but you could give the acceleration.Below are some answers which are all correct.

1. a=70metres per second every 7 seconds
2. a=600 metres per second every minute
3. a=10 metres per second every second

Answer 3. is the preferred,the easiest to work with and generally accepted way and is found by dividing the change of velocity/speed by the time in seconds.

In summary when it is stated that the acceleration is 10 metres per second squared(which is the same as 10metres per second per second ) what it means is that every second the increase in speed/velocity is 10m/s.

5. Feb 25, 2012

The actual meaning of m/s2 is (m/s)/s.It is simplified for easy writing notation.As mentioned above,every quantity need not have a "physical" meaning.Force's unit is written as
kg m2/s2.For this,we cannot write it as kg ((m/s)/s),this would just be non simplified and difficult to use.Pressure=Force/area,which is written as kg/ms2.You cannot visualize these quantities based on their units,but on their practical application,you can.

6. Feb 25, 2012

### nasu

The unit can be "simplified" this way because we the same units (seconds) in both places.
It is not necessary to be like this. We could use (km/h)/s to express mathematically "it goes from 0 to 100 km/h in 5 seconds".
Then you can "simplify" the (km/h)/s to km/(h.s). Would you question this "simplification" too?

7. Feb 25, 2012

### xzardaz

I think this is a terrible explanation. Maybe it is true ( you know better - I'm just a novice ) , but very ugly. But can't we just look for a reason ?

Can't you just use that anyway ?

Because 1 minute is 60 seconds you can convert :

( one meter per one minute ) per one minute = $\frac{\frac{m}{min}}{min}$ = $\frac{\frac{m}{60*s}}{60*s}$ = $\frac{\frac{1}{60}*\frac{m}{s}}{60*s}$ ( i think we can pull out 60 , because it is nothing but a number ) = $\frac{1}{60}*\frac{\frac{m}{s}}{60*s}$ = $\frac{1}{60}*\frac{1}{60}*\frac{\frac{m}{s}}{s}$ = $\frac{1}{60^2}*\frac{\frac{m}{s}}{s}$ = $\frac{\left(\frac{\frac{m}{s}}{s}\right)}{60^2}$

So the rule remains : if we have x meters per minute per minute , it is equal to x/602 meters per second per second , whithout having to have soconds2. We use the math laws for the coefficient 60 , not for the physics units of time and because the meaning of coefficients is to multiply the things before next to them and thereforе , they have to be able to obey the simple arithmetic laws. So 602 makes perfect sence. I just don't get the sense of s2.

For example , we can play with a speed ( to be simpler than acceleration ) :

$\frac{m}{s}$ = m*$\frac{1}{s}$.

Meters is distance , 1/seconds ( per second - Herz ) is frequancy.It is distance times a frequancy. It can be translated in english like that " the unit for speed is how frequent a point travels some distance ". In Si distance is mesured in meters , frequancy - in herz , so the speed can be mesured in meters by herz or herzes by meters. It makes perfect sense.

But I don't see the meaning of seconds2. Maybe it have some logic , but I don't get it. What can be mesured in seconds squared ?

It must be some square ( geometric ) with side one second in a diagram , which involves time and something else.

We can examine a moving ball , which changes its position in one dimension by the law x=t2. x is the distance of the ball from the origin ( the point where the ball began to move ) and t is the elapsed time.

We can draw something like this :

It is easyer to imagine it , when we add second dimension to the image , representing time. The position will look like a curve ( the square function ) in this space-time :

The speed of the ball is changing over time. So we can imagine one dimension , representing the speed. Imagine the speed is another ball , which travels in this dimension. I think the equation would be speed = 2*t. So we can draw similar picture like the first :

And again we can imagine space-time picture for the speed. This time the curve is representing the speed law. Here it is :

Again , we ca do this with the acceleration. We can see that in this case it stays the same ofer time ( but it could change over time , depending on the law of motion of the ball ). Here is the picture :

So there must be some geometric square in the above pictures , representing the seconds2. I just don't see it.

Another thing to view it is to imagine some 3 dimentional space - 1 dimension for time , 1 for speed and one for acceleration , and examine the figure. But I don't thing this will work , since the acceleration is the change of speed over time - one dimension will depend on the other.

We can also represent the above pictures changed - the axes won't be straight lines with right angles between them. For example the Speed dimention in the 4th picture will be the same as the square curve on the second picture. And because of the change of the coordinate system the speed curve (v=2*t) will no longer be a straight line. But I don't thing we can see any square here , also.

If we can see some square in the pictures , maybe it will give us a clue of the meaning of s2 and what it mesures.

edit :

As I wrote this , I did't saw the rest of the replies. I'm sorry. It makes sense to use m/s^2 for notation , still meaning m/s/s.

Last edited: Feb 25, 2012
8. Feb 25, 2012

### Michael C

You find this idea ugly. Why? We might say that $\frac{\frac{m}{s}}{s}$ is uglier than $\frac{m}{s^{2}}$. It's certainly more cumbersome. You could go on using $\frac{\frac{m}{s}}{s}$ but the calculations will be easier if you use $\frac{m}{s^{2}}$. The two expressions are mathematically identical, so we know that we can replace one with the other.

Have a look at the list of SI derived units on this page. You'll see all sorts of expressions with components that nobody can visualise. A farad, for instance (the unit of capacitance), contains squared amperes and seconds to the power of four. What is the physical meaning of a squared ampere, or a second to the power of four?

Last edited: Feb 25, 2012
9. Feb 25, 2012

Let me expand upon my answer above.When we describe that the acceleration of an object is,for example,7 metres per second squared (7 m/s^2) all that means is that every second the object gains in speed by 7 metres per second.calling it 7 metres per second per second is the same thing.

We can mix and match units and measure acceleration,for example,in miles per hour per minute.Such units,though correct,are not usually used.

10. Feb 25, 2012

### tiny-tim

actually, there are units which are confusing in this way

eg the resistivity, ρ, of a material

(also conductivity, permittivity, permeability)

(by "a material", i mean eg copper, as opposed to a particular copper wire)

ρ is measured in ohm-metres (Ω-m),

but it's really Ω-m2 per m …

because resistance R = ρl/A (l = length of the wire, A = cross-section area)

calling it Ω-m is confusing!

however, the combining of length and area (in resistivity) is different to the second-squared in acceleration …

in resistivity, we can change l and A separately (the are two different measurements), so arguably they deserve to be mentioned separately in the units

but in acceleration, the second-squared only involves one measurement, so there's no confusion

11. Feb 25, 2012