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i read Landau's book recently. In this book p.43

Landau says from (16.1) (16.2)

can be write down $T_10$= $p_0^2$/2$m_1$=($M-m_1$)($E_i-E_1i-E_i'$)/$M$

For me, it is hard to understand the factor ($M-m_1$)/$M$.

Can someone give me a hand to let me understanding master's idea?

If you can give me a easy example, i will appreciate you.

thx for your help.

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  • $\begingroup$ Could you clarify what all the T's and M's represent? $\endgroup$ Feb 20, 2014 at 20:39

1 Answer 1

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The result is straight forward. As Landau and Lifshitz explain in p.41, when a body disintegrates into two pieces of masses $m_1$ and $m_2$ respectively, their momenta must be equal in magnitude and oppositely directed by the law of conservation of momentum. So, let each body have momentum $p_0$. Then, $(16.1)$ and $(16.2)$ say that the difference in the internal energies equals the kinetic energy of the reduced mass.

$E_i - E_{1i} - E_{2i} = \frac{p_0 ^2}{2m}$ where $\frac{1}{m} = \frac{1}{m_1} + \frac{1}{m_2}$

Let us now see what happens when the body splits into more than two parts. Of course, we cannot say anything about the magnitudes and directions of the momentum of each body. But, we shall try to obtain some information about maximum possible kinetic energy and so on.

So, now we have one part with mass $m_1$ that we are interested in. If we sum the momenta of the remaining parts, we can conclude that the total momentum of the remaining parts is equal in magnitude and directed oppositely to the momentum of $m_1$ just like the previous case. Let it be $p_0$. Hence, we can now use the same equations for masses $m_1$ and $M-m_1$

$\frac{p_0 ^2}{2} (\frac{1}{m_1} + \frac{1}{M-m_1}) = E_i - E_{1i} - E_{i}^{'}$ which on simplification gives

$\frac{p_0 ^2}{2m_1} \frac{M}{M - m_1} = E_i - E_{1i} - E_{i}^{'}$

and the result follows.

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