Please explain what is wrong with my relativistic momentum problem

[Virous]

Hello!

First, please, zoom in the attached file and take a look on it. Image 1 demonstrates the situation: two identical balls collide, horizontal component of velocity of both remains unchanged and vertical component becomes opposite as a result of the collision. Table in image 3 shows all the velocities in terms of a (horizontal component) and b (vertical).

Now we change the frame of reference into the one, which moves to the right on the speed of a. It is demonstrated on Image 2. New velocities are calculated via the relativistic velocity addition formula and are shown in the table on image 4.

Finally I calculated total momentum before (image 5) and total momentum after (image 6). Results are not equal! What did I do wrong?

http://stg704.rusfolder.com/preview/20140518/7/40730757_3_1176191.png

Deep thanks!

 

[xox]

Hello!

First, please, zoom in the attached file and take a look on it. Image 1 demonstrates the situation: two identical balls collide, horizontal component of velocity of both remains unchanged and vertical component becomes opposite as a result of the collision. Table in image 3 shows all the velocities in terms of a (horizontal component) and b (vertical).

Now we change the frame of reference into the one, which moves to the right on the speed of a. It is demonstrated on Image 2. New velocities are calculated via the relativistic velocity addition formula and are shown in the table on image 4.

Finally I calculated total momentum before (image 5) and total momentum after (image 6). Results are not equal! What did I do wrong?

http://stg704.rusfolder.com/preview/20140518/7/40730757_3_1176191.png

Deep thanks!

Relativistic momentum is $\vec{p}=\gamma(v)m \vec{v}$. You seem to be missing the "gammas" in your formulas.

 

[Virous]

No, I don`t. On images 5-6 y - is gamma with respect to a.

Other gammas (with respect to other things) are already expanded.

 

[mfb]

The velocities of the balls won't be the same in the new reference frame, so the betas are different for the different particles.

It would help to see how you derived those expressions for the momenta.

 

[xox]

No, I don`t. On images 5-6 y - is gamma with respect to a.

Other gammas (with respect to other things) are already expanded.

OK, try writing by components:

$$\gamma(v_A) \vec{v_A}+\gamma(v_B) \vec{v_B}=\gamma(v'_A) \vec{v'_A}+\gamma(v'_B) \vec{v'_B}$$

It is difficult to decipher from your formulas.

 

[Virous]

True, I`m sorry. I have to expand a little bit more.

mfb, I did :)

So:

Hopefully the table on Image 4 is clear. It is just the velocity addition formula as follows:

http://stg660.rusfolder.com/preview/20140518/2/40731152_3_1176206.jpg

I used the "x" one for "a" calculations, since "a" is parallel to frame motion and "y" one for "b" calculations since it is perpendicular. Anyway, no doubt in that table, since it is given in the book :)

Now, to calculate momentum used this one:

http://stg741.rusfolder.com/preview/20140518/2/40731162_3_1176207.jpg

Then a simple substitution of what was in the table instead of velocity there :)

 

[Virous]

And, of course, I summed up initial momenta of balls on image 5 and finals of image 6

 

[Meir Achuz]

You have to use conservation of momentum to find the final velocities in either case.
The only way you can violate conservation of momentum is by making an error..

 

[Virous]

No, I`m doing an opposite thing. I took an experimental fact, which is the case, and want to make sure, that my momentum is conserved. The book says, that it is so streight forward, so they let me to do this on my own. I spent a day, but it is not conserved :)

That is why I`m here trying to figure out, what did I do wrong.

 

[dauto]

what are you using for the γ and β in the expression for the velocities?

 

[xox]

True, I`m sorry. I have to expand a little bit more.

mfb, I did :)

So:

Hopefully the table on Image 4 is clear. It is just the velocity addition formula as follows:

http://stg660.rusfolder.com/preview/20140518/2/40731152_3_1176206.jpg

I used the "x" one for "a" calculations, since "a" is parallel to frame motion and "y" one for "b" calculations since it is perpendicular. Anyway, no doubt in that table, since it is given in the book :)

Now, to calculate momentum used this one:

http://stg741.rusfolder.com/preview/20140518/2/40731162_3_1176207.jpg

Then a simple substitution of what was in the table instead of velocity there :)

But you seem to be using the SAME $\gamma$ in all you formulas , this is not right. The " gammas" must be ALL DIFFERENT, see post 5. It is difficult to see what you are doing, I asked you two write the formulas component by component (first x, then y). Then we can see what is going wrong. Can you do it for x first?

 

[Virous]

x has no problems. It is conserved. I do not include it at all. Working only with y.

dauto, I`m not. Other gammas are expanded.

 

[xox]

x has no problems. It is conserved.

ok

Working only with y.

...but you seem to have the SAME "gamma" in all the terms (it is difficult to decipher your writeup, so I might be wrong).

It is also not clear what you have as equation as conservation, there is no equation to speak of, where is your:

$$\gamma(v_A) \vec{v_A}+\gamma(v_B) \vec{v_B}=\gamma(v'_A) \vec{v'_A}+\gamma(v'_B) \vec{v'_B}$$

 

[Virous]

Yes, I do, but it comes from relativistic velocity addition, as I wrote before. It must be the same, since we are in the same frame. Again, it is given in the book. Then I`m using the second formula (momentum one) and substitute data from the table, getting different gammas.

 

[xox]

Yes, I do, but it comes from relativistic velocity addition, as I wrote before. It must be the same, since we are in the same frame. Again, it is given in the book. Then I`m using the second formula (momentum one) and substitute data from the table, getting different gammas.

There are FIVE "gammas": $\gamma(v_A),\gamma(v_B), \gamma(v'_A),\gamma(v'_B), \gamma(V)$

I do not see them but this might be due to the way you write your formulas. Can you post them in LaTeX?

 

[Virous]

Yes, I`ll try to :) I have to leave now. In approximately 8 hours I`ll be back and take one more try! Thank you for staying with me :)

 

[xox]

Yes, I`ll try to :) I have to leave now. In approximately 8 hours I`ll be back and take one more try! Thank you for staying with me :)

I found the mistake, it was difficult due to the presentation. Looking at image 5:

You calculate $\gamma(v_A)$ correctly but your $\gamma(v_B)$ is incorrect, you need to use $v_B^2=(\frac{2a}{1+\beta^2})^2+(\frac{b}{\gamma(1+\beta^2})^2$

You have used only the y -component of $v_B$, you completely forgot to add the x component.

 

[Virous]

Even earlier :) So, we start from the very beginning once again step by step.

Image 1 shows a situation: collision of two balls. This comes from an experimental fact, so it must be true and the momentum must be conserved. Initial and final velocities of two balls are given in the table on image 3. Here a represents horizontal and b - vertical components.

Now if one computes the momenta of each of the balls before and after, one will find, that it is conserved in both classical ($p = mu$) and relativistic ($p= \gamma mu$) mechanics. It is simple.

Now we move to another frame, which moves with a constant velocity a to the right with respect to the first frame. It moves in such a way, that in classical mechanics the Ball 1 will have 0 horizontal velocity, and the Ball 2 will have a horizontal velocity of -2a. Vertical velocities are the same.

In relativistic mechanics it is more complicated and we have to use relativistic velocity addition formula to get all the velocities listed in the table on image 4 (in the brackets the first equation is a horizontal and the second is a vertical components).

Calculation of momentum for horizontal components shows, that it is conserved. We are not interested in it. Now the vertical ones:

$p= \gamma mu$

Total momenta before the collision:

$\sum p_{before}=\gamma(u_{1}) mu_{1}+\gamma(u_{2}) mu_{2}=\frac{mu_{1}}{\sqrt{1-(\frac{u_{1}^2}{c^2})}}+\frac{mu_{2}}{\sqrt{1-(\frac{u_{2}^2}{c^2})}}$

That is how I got my equations. Then we just substitute the data from the table instead of u1 and u2 and do the same for total momenta after.

And then we find, that now the momentum is not conserved. That`s the problem!

 

[Virous]

xox, firstly, I`m not interested in horizontals, since they are conserved as they are (checked that). But I tried to add them to the equation as well. It makes no difference. Momentum is still not conserved.

 

[PAllen]

This may be equivalent to what xox is saying, but you cannot use the co-linear velocity addition formula:

(u + v) / (1 + uv/c²)

for any non-colinear situation. Further, you cannot use it component wise in such a situation. There are several ways to state a more general velocity additions formula. My favorite is:

γ(relative) = γ(u)γ(v)(1-uv cos(θ))

then, from relative gamma you get relative speed.

 

[Virous]

Offf...

I can, just a formula is a little bit different. Again, I added velocities correctly, because it is stated in the book.

In your formulas, PAllen, when θ=90, the formula turns into my one for verticals, and when it is 0 into my one for horizontals:

http://stg660.rusfolder.com/preview/20140518/2/40731152_3_1176206.jpg

 

[PAllen]

In relativistic mechanics it is more complicated and we have to use relativistic velocity addition formula to get all the velocities listed in the table on image 4 (in the brackets the first equation is a horizontal and the second is a vertical components).

And this is the problem. The velocity addition formula you use is wrong for this situation. It only applies to colinear velocities, and cannot be used component-wise.

Much simpler (for this problem), actually, is to arrive at values for S' by Lorentz transform. Then you don't need to worry about general velocity addition formulat - it is all contained in the Lorentz transform, from which all such auxiliary formulas are derived.

 

[Virous]

If there is a mistake it is either in the nature (less probable) or in the last step. Velocity addition is correct definitely :)

 

[Virous]

And this is the problem.

This is not :) I used two formulas: for col-linear and perpendicular velocities where appropriate.

 

[Virous]

http://stg660.rusfolder.com/preview/20140518/2/40731152_3_1176206.jpg

Left one works for col-linear. Right one for perpendicular. They both can be derived from your one with θ.

 

[PAllen]

This is not :) I used two formulas: for col-linear and perpendicular velocities where appropriate.

But you cannot break down the components that way. To get velocity of the 'top' ball in S' (for example), you must use the total formula with an angle neither 0 nor 90. It is not valid to break it up into colinear and orthogonal components.

 

[xox]

xox, firstly, I`m not interested in horizontals, since they are conserved as they are (checked that). But I tried to add them to the equation as well. It makes no difference. Momentum is still not conserved.

I don't think you understand, you need the value $v_B^2$ in the calculation for $\gamma(v_B)$, you are incorrectly using only the y component. You make the same mistake in calculating $\gamma(v'_B)$

 

[Virous]

PAllen It is correct, since it is what was in the book. The mistake is in the second step. Just forget about horizontal component existence.

 

[Virous]

xox, please, write down the correct total momentum before and after equations, so I will understand it better :)

 

[PAllen]

PAllen It is correct, since it is what was in the book. The mistake is in the second step. Just forget about horizontal component existence.

For momenta, you can treat components separately. For relative velocities, you cannot. As to what is in the book, I don't have the book and thus it is more likely (to me) that you misinterpret what is stated in the book.

 

[Virous]

I just copied the table as it is. It is not mine. And I don`t see any mistakes in it.

 

[xox]

xox, please, write down the correct total momentum before and after equations, so I will understand it better :)

Before: $$\gamma(v_A) \vec{v_A}+\gamma(v_B) \vec{v_B}$$


After: $$\gamma(v'_A) \vec{v'_A}+\gamma(v'_B) \vec{v'_B}$$

You have everything correct, except the way you calculate $\gamma(v_B), \gamma(v'_B)$.
I explained the error.

 

[Virous]

No, I mean, please, expand it (gammas). And it seems your forgot masses.

 

[xox]

For momenta, you can treat components separately. For relative velocities, you cannot. As to what is in the book, I don't have the book and thus it is more likely (to me) that you misinterpret what is stated in the book.

He's projecting

$$\gamma(v_A) \vec{v_A}+\gamma(v_B) \vec{v_B}=\gamma(v'_A) \vec{v'_A}+\gamma(v'_B) \vec{v'_B}$$

on the x and y axis, so his using of the components is correct. His error is in calculating the $\gamma(v_B),\gamma(v'_B)$

 

[xox]

No, I mean, please, expand it (gammas). And it seems your forgot masses.

The masses cancel out because both particles have the same rest mass.