Новое решение

astrosander правка от 1 апр. 2025 г., 01:33

правка #13989 позже →

en/14.5.6.md +52 −0

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		+	### Statement
		+
		+	$14.5.6.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	### 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
		+
		+	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:
		+
		+	$$
		+	M_1 = M + m
		+	$$
		+
		+	#### Answer
		+
		+	\[
		+	M_1 = M + m \\
		+	p = mc
		+	\]
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

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			### Statement

			$14.5.6.$ [Insert the problem statement]

			### Solution

			### 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

			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:

			$$
			M_1 = M + m
			$$

			#### Answer

			\[
			M_1 = M + m \\
			p = mc
			\]

			#### Answer

			[Insert a concise answer or boxed result]