Правка разделов «Problem», «Answer»
en/14.5.6.md
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| − | ### Statement | ||
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| − | $14.5.6.$ [Insert the problem statement] | ||
| − | |||
| − | ### Solution | ||
| − | |||
| ### Problem | |||
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| + | $14.5.6.$ A stationary atom of mass \( M \) absorbs a photon of mass \( m \). Determine the mass and momentum of the atom after the photon is absorbed. | ||
| ### Solution | |||
| A photon has momentum \( p_{\gamma} = mc \), and after absorption, this momentum must be conserved. Let the atom move with momentum \( p \) after absorbing the photon, then: | |||
| $$ | |||
| p = mc | |||
| $$ | |||
| The initial energy of the system is the rest energy of the atom plus the energy of the photon: | |||
| $$ | |||
| E_{\text{initial}} = Mc^2 + mc^2 | |||
| $$ | |||
| After absorption, the atom will have a new mass \( M_1 \) and kinetic energy: | |||
| $$ | |||
| E_{\text{final}} = M_1 c^2 | |||
| $$ | |||
| Since total energy must be conserved: | |||
| $$ | |||
| Mc^2 + mc^2 = M_1 c^2 | |||
| $$ | |||
| Solving for the new mass of the atom: | |||
| $$ | |||
| M_1 = M + m | |||
| $$ | |||
| @@ -43,10 +37,9 @@Solution | |||
| #### Answer | |||
| \[ | |||
| − | M_1 = M + m | ||
| − | p = mc | ||
| + | M_1 = M + m | ||
| \] | |||
| − | #### Answer | ||
| − | |||
| − | [Insert a concise answer or boxed result] | ||
| + | \[ | ||
| + | p = mc | ||
| + | \] | ||
| @@ -1,12 +1,6 @@ | |||
| ### Statement | |||
| $14.5.6.$ [Insert the problem statement] | |||
| ### Solution | |||
| ### Problem | ### Problem | ||
| $14.5.6.$ A stationary atom of mass \( M \) absorbs a photon of mass \( m \). Determine the mass and momentum of the atom after the photon is absorbed. | |||
| ### Solution | ### Solution | ||
| A photon has momentum \( p_{\gamma} = mc \), and after absorption, this momentum must be conserved. Let the atom move with momentum \( p \) after absorbing the photon, then: | A photon has momentum \( p_{\gamma} = mc \), and after absorption, this momentum must be conserved. Let the atom move with momentum \( p \) after absorbing the photon, then: | ||
| $$ | $$ | ||
| p = mc | p = mc | ||
| $$ | $$ | ||
| The initial energy of the system is the rest energy of the atom plus the energy of the photon: | The initial energy of the system is the rest energy of the atom plus the energy of the photon: | ||
| $$ | $$ | ||
| E_{\text{initial}} = Mc^2 + mc^2 | E_{\text{initial}} = Mc^2 + mc^2 | ||
| $$ | $$ | ||
| After absorption, the atom will have a new mass \( M_1 \) and kinetic energy: | After absorption, the atom will have a new mass \( M_1 \) and kinetic energy: | ||
| $$ | $$ | ||
| E_{\text{final}} = M_1 c^2 | E_{\text{final}} = M_1 c^2 | ||
| $$ | $$ | ||
| Since total energy must be conserved: | Since total energy must be conserved: | ||
| $$ | $$ | ||
| Mc^2 + mc^2 = M_1 c^2 | Mc^2 + mc^2 = M_1 c^2 | ||
| $$ | $$ | ||
| Solving for the new mass of the atom: | Solving for the new mass of the atom: | ||
| $$ | $$ | ||
| M_1 = M + m | M_1 = M + m | ||
| $$ | $$ | ||
| @@ -43,10 +37,9 @@Solution | |||
| #### Answer | #### Answer | ||
| \[ | \[ | ||
| M_1 = M + m |
M_1 = M + m | ||
| p = mc | |||
| \] | \] | ||
| #### Answer | \[ | ||
| p = mc | |||
| [Insert a concise answer or boxed result] | \] | ||