Post by **LinearSpoon** » 10 Mar 2014, 16:20

@

**trismarck**, I'm afraid I don't understand your reference, but I do realize there is a joke there.

@

**Sjc1000**, regarding the car & headlights question: If you believe Einstein's theory of special relativity, the answer is A. There's no reason you shouldn't believe the theory, it has been experimentally verified multiple times and even has practical uses, for example in GPS calculations.

The velocity vector addition that we typically use is a simplification of the true formula which works because we tend to deal with speeds that are a small fraction of the speed of light.

Here is Einstein's formula for adding relative velocities

a and

b to find their total velocity,

v relative to the observer.

c is the speed of light, which is constant for all observers.

You can see that if

a and

b are small numbers, the denominator of this formula may as well be 1+0.

Lets analyze this situation. Let

a be the velocity of the car, and

b the velocity of the light from the headlights. We will choose the driver of the car as the observer.

Relative to the driver, the car is moving 0 m/s. Relative to the driver, the light will exit the headlights at

c m/s.

So:

Code: Select all

```
a = 0
b = c
0 + c
v = -----------------
1 + (0*c / c^2)
c
v = ------- = c
1 + 0
```

The driver will observe the light travelling forward from his car at

c, or 299792458 m/s, a number which is exact because the meter is defined based on the speed of light.

As an interesting side point, you can also perform the same calculation as a stationary observer.

The car is moving at

c m/s relative to the stationary observer, and the light comes out of the car headlight at

c m/s.

Code: Select all

```
a = c
b = c
c + c
v = -----------------
1 + (c*c / c^2)
2*c
v = ------- = c
1 + 1
```

So the stationary observer also sees the headlight moving at 299792458 m/s relative to himself.

The reason this works out to the same speed is because time passes differently for each observer, and space "shrinks" when observing objects that have a different velocity than your own. This is where people are getting that it is possible to time travel by moving near the speed of light. It should also be noted that this is purely theoretical, it's impossible for any object with mass to reach the speed of light. This is because as you reach higher velocities, your mass will increase, which will end up requiring an infinite amount of force to continue accelerating towards the speed of light.

Some relevant Wikipedia links:

http://en.wikipedia.org/wiki/Velocity-addition_formula
http://en.wikipedia.org/wiki/Length_contraction
http://en.wikipedia.org/wiki/Time_dilation
http://en.wikipedia.org/wiki/Hafele%E2% ... experiment
And a simple calculator.

a and

b are velocities relative to an observer,

Classic Physics returns the common velocity addition,

Special Theory returns Einstein's velocity addition. All units are m/s.

Code: Select all

```
Gui, Add, Text, x12 y9 w100 h30 , a (m/s)
Gui, Add, Text, x12 y39 w100 h30 , b (m/s)
Gui, Add, Text, x12 y69 w100 h30 , Classic Physics`n a+b
Gui, Add, Text, x12 y99 w100 h30 , Special Theory`n (a+b)/(1+(ab/c^2))
Gui, Add, Edit, x122 y9 w150 h20 va gUpdate, 0
Gui, Add, Edit, x122 y39 w150 h20 vb gUpdate, 0
Gui, Add, Edit, x122 y69 w150 h20 ReadOnly vClassic
Gui, Add, Edit, x122 y99 w150 h20 ReadOnly vSpecial
Gui, Show
c := 299792458 ;speed of light in m/s
Update:
Gui, Submit, NoHide
GuiControl,, Classic, % a+b
GuiControl,, Special, % (a+b)/(1+(a*b/c**2))
return
GuiClose:
ExitApp
return
```

Some interesting things to try:

1. Very low values for a and b.

2. Values of 500,000 for each.

3. The speed of light (299792458) for a and any other value for b.

I hope that explanation wasn't too detailed. I think I got carried away researching it..fascinating stuff though.

@[b]trismarck[/b], I'm afraid I don't understand your reference, but I do realize there is a joke there.

@[b]Sjc1000[/b], regarding the car & headlights question: If you believe Einstein's theory of special relativity, the answer is A. There's no reason you shouldn't believe the theory, it has been experimentally verified multiple times and even has practical uses, for example in GPS calculations.

The velocity vector addition that we typically use is a simplification of the true formula which works because we tend to deal with speeds that are a small fraction of the speed of light.

Here is Einstein's formula for adding relative velocities [c]a[/c] and [c]b[/c] to find their total velocity, [c]v[/c] relative to the observer. [c]c[/c] is the speed of light, which is constant for all observers.

[img]http://i.imgur.com/ba9rU4W.png[/img]

You can see that if [c]a[/c] and [c]b[/c] are small numbers, the denominator of this formula may as well be 1+0.

Lets analyze this situation. Let [c]a[/c] be the velocity of the car, and [c]b[/c] the velocity of the light from the headlights. We will choose the driver of the car as the observer.

Relative to the driver, the car is moving 0 m/s. Relative to the driver, the light will exit the headlights at [c]c[/c] m/s.

So:

[code]a = 0

b = c

0 + c

v = -----------------

1 + (0*c / c^2)

c

v = ------- = c

1 + 0[/code]The driver will observe the light travelling forward from his car at [c]c[/c], or 299792458 m/s, a number which is exact because the meter is defined based on the speed of light.

As an interesting side point, you can also perform the same calculation as a stationary observer.

The car is moving at [c]c[/c] m/s relative to the stationary observer, and the light comes out of the car headlight at [c]c[/c] m/s.

[code]a = c

b = c

c + c

v = -----------------

1 + (c*c / c^2)

2*c

v = ------- = c

1 + 1[/code]So the stationary observer also sees the headlight moving at 299792458 m/s relative to himself.

The reason this works out to the same speed is because time passes differently for each observer, and space "shrinks" when observing objects that have a different velocity than your own. This is where people are getting that it is possible to time travel by moving near the speed of light. It should also be noted that this is purely theoretical, it's impossible for any object with mass to reach the speed of light. This is because as you reach higher velocities, your mass will increase, which will end up requiring an infinite amount of force to continue accelerating towards the speed of light.

Some relevant Wikipedia links:

[url]http://en.wikipedia.org/wiki/Velocity-addition_formula[/url]

[url]http://en.wikipedia.org/wiki/Length_contraction[/url]

[url]http://en.wikipedia.org/wiki/Time_dilation[/url]

[url]http://en.wikipedia.org/wiki/Hafele%E2%80%93Keating_experiment[/url]

And a simple calculator. [c]a[/c] and [c]b[/c] are velocities relative to an observer, [c]Classic Physics[/c] returns the common velocity addition, [c]Special Theory[/c] returns Einstein's velocity addition. All units are m/s.

[code=autohotkey file=Script.ahk]Gui, Add, Text, x12 y9 w100 h30 , a (m/s)

Gui, Add, Text, x12 y39 w100 h30 , b (m/s)

Gui, Add, Text, x12 y69 w100 h30 , Classic Physics`n a+b

Gui, Add, Text, x12 y99 w100 h30 , Special Theory`n (a+b)/(1+(ab/c^2))

Gui, Add, Edit, x122 y9 w150 h20 va gUpdate, 0

Gui, Add, Edit, x122 y39 w150 h20 vb gUpdate, 0

Gui, Add, Edit, x122 y69 w150 h20 ReadOnly vClassic

Gui, Add, Edit, x122 y99 w150 h20 ReadOnly vSpecial

Gui, Show

c := 299792458 ;speed of light in m/s

Update:

Gui, Submit, NoHide

GuiControl,, Classic, % a+b

GuiControl,, Special, % (a+b)/(1+(a*b/c**2))

return

GuiClose:

ExitApp

return

[/code]Some interesting things to try:

1. Very low values for a and b.

[spoiler]Notice that both calculations are approximately the same (or even exactly the same, due to the imprecise nature of floating point numbers in computers). This is why we used the original calculation, because humans rarely if ever measured or dealt with high enough speeds for the difference to matter. This is also why the simpler formula is still widely used and taught in schools, because the difference won't matter for the average person.[/spoiler]

2. Values of 500,000 for each.

[spoiler]Here you should start to see the results for Classic vs Special drift apart. Even still, the difference is so small it could easily be mistaken for a measurement error...This is why we didn't really notice a problem until the late 1800s when Maxwell's equations were found to conflict with classical motion mechanics.[/spoiler]

3. The speed of light (299792458) for a and any other value for b.

[spoiler]This is where people are getting that the speed of light is the "speed limit" of the universe...This is also where the Classic formula is most lacking - in prediction of very high speed objects.[/spoiler]

I hope that explanation wasn't too detailed. I think I got carried away researching it..fascinating stuff though.