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## Main Question or Discussion Point

Hi

I checked this problem many times but I didn't end up with the result wanted.

Assume that a particle of rest mass [tex]m_0[/tex], (relativistic) energy [tex]e_0[/tex] and (relativistic) momentum [tex]p_0[/tex] is moving in a straight line. This particle suddenly hits a stationary particle with rest mass [tex]M_0[/tex] ahead and they both get involved in an elastic collision. As a result of the collision, the second particle gains momentum [tex]P[/tex] and energy [tex]E[/tex] and the first one keeps moving with a new momentum, [tex]p[/tex], while its energy is now [tex]e.[/tex]

In the Newtonian limit, using the conservation laws of energy and momentum we can get

[tex]P=\frac{2p_0M_0}{M_0+m_0},[/tex]

[tex]p=\frac{p_0(m_0-M_0)}{M_0+m_0}.[/tex]

But in the relativistic case, the two conservation laws get really sloppy and complicated, though it is claimed that, for example,

[tex]P=\frac{2p_0M_0(e_0+M_0c^2)}{2M_0e_0+m^2_0c^2+M^{2}_0c^2}.[/tex]

How can we obtain this expression? Is this even correct?

Thanks in advance

AB

I checked this problem many times but I didn't end up with the result wanted.

Assume that a particle of rest mass [tex]m_0[/tex], (relativistic) energy [tex]e_0[/tex] and (relativistic) momentum [tex]p_0[/tex] is moving in a straight line. This particle suddenly hits a stationary particle with rest mass [tex]M_0[/tex] ahead and they both get involved in an elastic collision. As a result of the collision, the second particle gains momentum [tex]P[/tex] and energy [tex]E[/tex] and the first one keeps moving with a new momentum, [tex]p[/tex], while its energy is now [tex]e.[/tex]

In the Newtonian limit, using the conservation laws of energy and momentum we can get

[tex]P=\frac{2p_0M_0}{M_0+m_0},[/tex]

[tex]p=\frac{p_0(m_0-M_0)}{M_0+m_0}.[/tex]

But in the relativistic case, the two conservation laws get really sloppy and complicated, though it is claimed that, for example,

[tex]P=\frac{2p_0M_0(e_0+M_0c^2)}{2M_0e_0+m^2_0c^2+M^{2}_0c^2}.[/tex]

How can we obtain this expression? Is this even correct?

Thanks in advance

AB

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