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		−	~~<h2>~~Solutions of Savchenko Problems in Physics~~</h2~~>
		−	~~<p class="author"~~>
		−	Aliaksandr Melnichenka <br/>
		−	October 2023
		−	</p>
		+	<div id = "logo">
		+	<span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span>
		+	</div>
		+	<p class="author">
		+	Solutions of Savchenko Problems in Physics <br/>
		+	knowledge must be free
		+	</p>
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			<h3 id="back-link"><a href="../../#2.2">$\leftarrow$Back</a></h3>

			<h3> Statement </h3>

			<p>
			$2.2.16$
			The radioactive core decayed into three fragments of mass $m_1$, $m_2$, and $m_3$, having mutually perpendicular velocities $v_1$, $v_2$, and $v_3$, respectively. What was the speed of the nucleus before the decay?
			</p>
			<h3>Solution</h3>
			<p>
			Let's consider the following figure
			</p>
			<center>
			<figure>
			<img src="draw.png"
			loading="lazy" alt="2.2.16" width="200" />
			<figcaption>
			Velocity's analysis
			</figcaption>
			</figure>
			</center>
			<p>
			From Momentum Conservation Law (MCL)
			$$(m_1+m_2+m_3)~\vec{V} = m_1\vec{v_1}+m_2\vec{v_2}+m_3\vec{v_3}$$
			Since velocity's vectors are mutually perpendicular, their directions can be fundamental for establishing the spatial coordinated system. So, let be $\hat{v_1}$, $\hat{v_2}$ and $\hat{v_3}$ the unitary vectors for each direction. Applying MCL for each direction,
			$$(m_1+m_2+m_3)V_1 = m_1v_1 \;(1)$$
			$$(m_1+m_2+m_3)V_2 = m_2v_2 \;(2)$$
			$$(m_1+m_2+m_3)V_3 = m_3v_3 \;(3)$$
			where $V_1$, $V_2$ and $V_3$ are the components of velocity $\vec{V}$,
			$$V=\sqrt{V_1^2+V_2^2+V_3^2} \;(4)$$
			Substituting $(1)$, $(2)$ and $(3)$ into $(4)$
			$$\boxed{V=\frac{\sqrt{m_1^2v_1^2+m_2^2v_2^2+m_3^2v_3^2}}{m_1+m_2+m_3}}$$
			</p>
			<p style="text-align: right; font-style: italic; font-size: 14;">
			BSc. Luis Daniel Fernández Quintana<br>
			Physics Department (FCNE)<br>
			Universidad de Oriente, Cuba<br>
			</p>



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	<body style="">		<body style="">
	<header style="text-align:center;">		<header style="text-align:center;">
	~~<h2>~~Solutions of Savchenko Problems in Physics~~</h2~~>		<div id = "logo">
	~~<p class="author"~~>		<span><img src = "../../img/book.png"><span><span>Savchenko Solutions</span>
	Aliaksandr Melnichenka <br/>		</div>
	October 2023		<p class="author">
	</p>		Solutions of Savchenko Problems in Physics <br/>
			knowledge must be free
			</p>
	</header>		</header>

	<h3 id="back-link"><a href="../../#2.2">$\leftarrow$Back</a></h3>		<h3 id="back-link"><a href="../../#2.2">$\leftarrow$Back</a></h3>

	<h3> Statement </h3>		<h3> Statement </h3>

	<p>		<p>
	$2.2.16$		$2.2.16$
	The radioactive core decayed into three fragments of mass $m_1$, $m_2$, and $m_3$, having mutually perpendicular velocities $v_1$, $v_2$, and $v_3$, respectively. What was the speed of the nucleus before the decay?		The radioactive core decayed into three fragments of mass $m_1$, $m_2$, and $m_3$, having mutually perpendicular velocities $v_1$, $v_2$, and $v_3$, respectively. What was the speed of the nucleus before the decay?
	</p>		</p>
	<h3>Solution</h3>		<h3>Solution</h3>
	<p>		<p>
	Let's consider the following figure		Let's consider the following figure
	</p>		</p>
	<center>		<center>
	<figure>		<figure>
	<img src="draw.png"		<img src="draw.png"
	loading="lazy" alt="2.2.16" width="200" />		loading="lazy" alt="2.2.16" width="200" />
	<figcaption>		<figcaption>
	Velocity's analysis		Velocity's analysis
	</figcaption>		</figcaption>
	</figure>		</figure>
	</center>		</center>
	<p>		<p>
	From Momentum Conservation Law (MCL)		From Momentum Conservation Law (MCL)
	$$(m_1+m_2+m_3)~\vec{V} = m_1\vec{v_1}+m_2\vec{v_2}+m_3\vec{v_3}$$		$$(m_1+m_2+m_3)~\vec{V} = m_1\vec{v_1}+m_2\vec{v_2}+m_3\vec{v_3}$$
	Since velocity's vectors are mutually perpendicular, their directions can be fundamental for establishing the spatial coordinated system. So, let be $\hat{v_1}$, $\hat{v_2}$ and $\hat{v_3}$ the unitary vectors for each direction. Applying MCL for each direction,		Since velocity's vectors are mutually perpendicular, their directions can be fundamental for establishing the spatial coordinated system. So, let be $\hat{v_1}$, $\hat{v_2}$ and $\hat{v_3}$ the unitary vectors for each direction. Applying MCL for each direction,
	$$(m_1+m_2+m_3)V_1 = m_1v_1 \;(1)$$		$$(m_1+m_2+m_3)V_1 = m_1v_1 \;(1)$$
	$$(m_1+m_2+m_3)V_2 = m_2v_2 \;(2)$$		$$(m_1+m_2+m_3)V_2 = m_2v_2 \;(2)$$
	$$(m_1+m_2+m_3)V_3 = m_3v_3 \;(3)$$		$$(m_1+m_2+m_3)V_3 = m_3v_3 \;(3)$$
	where $V_1$, $V_2$ and $V_3$ are the components of velocity $\vec{V}$,		where $V_1$, $V_2$ and $V_3$ are the components of velocity $\vec{V}$,
	$$V=\sqrt{V_1^2+V_2^2+V_3^2} \;(4)$$		$$V=\sqrt{V_1^2+V_2^2+V_3^2} \;(4)$$
	Substituting $(1)$, $(2)$ and $(3)$ into $(4)$		Substituting $(1)$, $(2)$ and $(3)$ into $(4)$
	$$\boxed{V=\frac{\sqrt{m_1^2v_1^2+m_2^2v_2^2+m_3^2v_3^2}}{m_1+m_2+m_3}}$$		$$\boxed{V=\frac{\sqrt{m_1^2v_1^2+m_2^2v_2^2+m_3^2v_3^2}}{m_1+m_2+m_3}}$$
	</p>		</p>
	<p style="text-align: right; font-style: italic; font-size: 14;">		<p style="text-align: right; font-style: italic; font-size: 14;">
	BSc. Luis Daniel Fernández Quintana<br>		BSc. Luis Daniel Fernández Quintana<br>
	Physics Department (FCNE)<br>		Physics Department (FCNE)<br>
	Universidad de Oriente, Cuba<br>		Universidad de Oriente, Cuba<br>
	</p>		</p>



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