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| <title>There is a bundle of identical nuclei moving with velocity v. The nuclei in the beam spontaneously divide into pairs of identical fragments. The velocity of the fragments moving in the direction of the beam is 3v. Find the velocity of the fragments moving in the direction perpendicular to the beam.</title> | | <title>There is a bundle of identical nuclei moving with velocity v. The nuclei in the beam spontaneously divide into pairs of identical fragments. The velocity of the fragments moving in the direction of the beam is 3v. Find the velocity of the fragments moving in the direction perpendicular to the beam.</title> |
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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
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| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.4.15^*.$ There is a bundle of identical nuclei moving with velocity $v$. The nuclei in the beam spontaneously divide into pairs of identical fragments. The velocity of the fragments moving in the direction of the beam is $3v$. Find the velocity of the fragments moving in the direction perpendicular to the beam. | | $1.4.15^*.$ There is a bundle of identical nuclei moving with velocity $v$. The nuclei in the beam spontaneously divide into pairs of identical fragments. The velocity of the fragments moving in the direction of the beam is $3v$. Find the velocity of the fragments moving in the direction perpendicular to the beam. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>Let's move to the beam center of mass frame.</p> | | <p>Let's move to the beam center of mass frame.</p> |
| <p>In this frame of reference, the relative velocity is related to the velocity in the NFR by the relation</p> | | <p>In this frame of reference, the relative velocity is related to the velocity in the NFR by the relation</p> |
| <p class="exp">$$ \vec{v'} = \vec{v} + \vec{v}_{rel} $$</p> | | <p class="exp">$$ \vec{v'} = \vec{v} + \vec{v}_{rel} $$</p> |
| <p>Where $\vec{v'}$ and $\vec{v}$ are the velocity in the inertial reference frame and the velocity of the reference frame of the system, respectively.</p> | | <p>Where $\vec{v'}$ and $\vec{v}$ are the velocity in the inertial reference frame and the velocity of the reference frame of the system, respectively.</p> |
| <p>By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$</p> | | <p>By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$</p> |
| | | |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.4.15/draw1.png" | | <img src="https://savchenkosolutions.com/1/1.4.15/draw1.png" |
| loading="lazy" width="175" /> | | loading="lazy" width="175" /> |
| <figcaption> | | <figcaption> |
| Representation of $\vec{v'}$ as a sum of two vectors | | Representation of $\vec{v'}$ as a sum of two vectors |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p>From where</p> | | <p>From where</p> |
| <p class="exp">$$v_{rel} = 2v$$</p> | | <p class="exp">$$v_{rel} = 2v$$</p> |
| <p>Next, let's consider the fragments that flew with the speed of $\vec{v}_\perp$)</p> | | <p>Next, let's consider the fragments that flew with the speed of $\vec{v}_\perp$)</p> |
| | | |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.4.15/draw.png" | | <img src="https://savchenkosolutions.com/1/1.4.15/draw.png" |
| loading="lazy" width="175" /> | | loading="lazy" width="175" /> |
| <figcaption> | | <figcaption> |
| Vector image | | Vector image |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p>Going back to inertial reference frame, we get that </p> | | <p>Going back to inertial reference frame, we get that </p> |
| <p class="exp">$$\vec{v}_\perp = \vec{v}_{rel}-\vec{v}$$</p> | | <p class="exp">$$\vec{v}_\perp = \vec{v}_{rel}-\vec{v}$$</p> |
| <p>By the Pythagorean theorem</p> | | <p>By the Pythagorean theorem</p> |
| <p class="exp">$$\fbox{${v}_\perp = 3{v}$}$$</p> | | <p class="exp">$$\fbox{${v}_\perp = 3{v}$}$$</p> |
| </p> | | </p> |
