Question about rearranging formula in Brian Cox's Why Does E=MC2

In summary, this has been driving me insane. I don't get how he went from s/c to t/y. If someone could explain step by step how you do it I would greatly appreciate it.
  • #1
DPR
7
0
This has been driving me insane. I don't get how he went from s/c to t/y. If someone could explain step by step how you do it I would greatly appreciate it.

Recall that we arrived at an expression for the length of the momentum vector in three-dimensional space, mΔx/Δt. We have just argued that Δx should be replaced by Δs and Δt should be replaced by Δs/c to form the four-dimensional momentum vector, which has a seemingly rather uninteresting length of mc. Indulge us for one more paragraph, and let us write the replacement for Δt, i.e., Δs/c, in full. Δs/c is equal to [sqrt (cΔt)^2)-(xΔ)^2]/c. This is a bit of a mouthful, but a little mathematical manipulation allows us to write it in a simpler form, i.e., it can also be written as Δt/γ where y=1/[sqrt 1-v^2/c^2)]. To obtain that, we have used the fact that υ = Δx/Δt is the speed of the object. Now γ is none other than the quantity we met in Chapter 3 that quantifies the amount by which time slows down from the point of view of someone observing a clock fly past at speed.
pg 127
 
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  • #2
could someone help me out with this or give me some tips on what to study to be able to figure this out?
 
  • #3
s is defined as [itex]s=\sqrt{-(\Delta x)^2+c^2(\Delta t)^2}[/itex]

Now divide both sides by [itex]c\Delta t[/itex] and you get:

[tex]\frac{s}{c\Delta t}=\sqrt{\frac{-(\Delta x)^2}{c^2(\Delta t)^2}+1}[/tex]

Now notice that [itex]\frac{\Delta x}{\Delta t}=v[/itex] to get:

[tex]\frac{s}{c\Delta t}=\sqrt{-\frac{v^2}{c^2}+1}[/tex]

Move the t to the right hand side and use the definition of gamma to get:
[tex]\frac{s}{c}=\frac{\Delta t}{\gamma}[/tex]
 
  • #4
hey thanks! I really need to brush up on my math!
 
  • #5
One additional point. We often call [itex]\frac{s}{c}[/itex] the proper time [itex]\tau=\frac{s}{c}[/itex]. You may see this proper time pop up more often depending on which sources you're reading.
 
  • #6
Matterwave said:
One additional point. We often call [itex]\frac{s}{c}[/itex] the proper time [itex]\tau=\frac{s}{c}[/itex]. You may see this proper time pop up more often depending on which sources you're reading.
Thanks. How exactly did you go do this step? Now divide both sides by [itex]c\Delta t[/itex] and you get:

[tex]\frac{s}{c\Delta t}=\sqrt{\frac{-(\Delta x)^2}{c^2(\Delta t)^2}+1}[/tex]

I get everything after that.
 
  • #7
DPR said:
Thanks. How exactly did you go do this step? Now divide both sides by [itex]c\Delta t[/itex] and you get:

[tex]\frac{s}{c\Delta t}=\sqrt{\frac{-(\Delta x)^2}{c^2(\Delta t)^2}+1}[/tex]

I get everything after that.

[tex]\frac{s}{c\Delta t}=\frac{\sqrt{-(\Delta x)^2+c^2(\Delta t)^2}}{\sqrt{(c\Delta t)^2}}=\sqrt{\frac{-(\Delta x)^2+c^2(\Delta t)^2}{c^2(\Delta t)^2}}=\sqrt{\frac{-(\Delta x)^2}{c^2(\Delta t)^2}+\frac{c^2(\Delta t)^2}{c^2(\Delta t)^2}}=\sqrt{\frac{-(\Delta x)^2}{c^2(\Delta t)^2}+1}[/tex]
 
  • #8
Just divide both sides like I said. You have to move the [itex]c\Delta t[/itex] inside the squareroot and divide both sides by it.

EDIT: What elfmotat said.
 
  • #9
thanks guys!
 

1. What is the purpose of rearranging formulas in Brian Cox's Why Does E=MC2?

The purpose of rearranging formulas is to simplify and manipulate equations in order to better understand the relationship between different variables and to solve for specific quantities.

2. How do you rearrange formulas in Brian Cox's Why Does E=MC2?

To rearrange formulas, you can use basic algebraic principles, such as moving terms to the other side of the equation, combining like terms, and using inverse operations to isolate the variable you are solving for.

3. Why is it important to rearrange formulas in Brian Cox's Why Does E=MC2?

Rearranging formulas allows for a deeper understanding of the concepts and principles behind the equation. It also allows scientists to make predictions and calculations based on the relationship between different variables.

4. Can you rearrange formulas in Brian Cox's Why Does E=MC2 for different variables?

Yes, you can rearrange the formula for any of the variables, including energy (E), mass (M), or the speed of light (C). The rearranged formula will show the relationship between the chosen variable and the other two variables.

5. Are there any limitations to rearranging formulas in Brian Cox's Why Does E=MC2?

The rearranged formula may not always provide a complete or accurate understanding of the concept, as other factors and principles may also affect the equation. Additionally, the rearranged formula is only applicable in certain scenarios and may not hold true in all situations.

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