Translated 1.4.1-1.4.18
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| + | <meta name="description" content="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."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="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."> | ||
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| + | <meta property="og:description" content="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."> | ||
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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> | ||
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| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <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. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <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 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>By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$</p> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.4.15/draw1.png" | ||
| + | loading="lazy" width="175" /> | ||
| + | <figcaption> | ||
| + | Representation of $\vec{v'}$ as a sum of two vectors | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p>From where</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> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.4.15/draw.png" | ||
| + | loading="lazy" width="175" /> | ||
| + | <figcaption> | ||
| + | Vector image | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p>Going back to inertial reference frame, we get that </p> | ||
| + | <p class="exp">$$\vec{v}_\perp = \vec{v}_{rel}-\vec{v}$$</p> | ||
| + | <p>By the Pythagorean theorem</p> | ||
| + | <p class="exp">$$\fbox{${v}_\perp = 3{v}$}$$</p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\sin \alpha = u/v$$ | ||
| + | </p> | ||
| + | |||
| + | |||
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| + | <br> | ||
| + | <p> | ||
| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
| + | </p> | ||
| + | <p> | ||
| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | ||
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| @@ -0,0 +1,105 @@ | |||
| <!DOCTYPE html> | |||
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| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <meta http-equiv="content-language" content="en"> | |||
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="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."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="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."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="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."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <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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| }); | |||
| </script> | |||
| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#1.4">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <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. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <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 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>By the condition, when $\vec{v}_{rel}$ and $\vec{v}$ are co-directed $v' = 3 v$</p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.4.15/draw1.png" | |||
| loading="lazy" width="175" /> | |||
| <figcaption> | |||
| Representation of $\vec{v'}$ as a sum of two vectors | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p>From where</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> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.4.15/draw.png" | |||
| loading="lazy" width="175" /> | |||
| <figcaption> | |||
| Vector image | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p>Going back to inertial reference frame, we get that </p> | |||
| <p class="exp">$$\vec{v}_\perp = \vec{v}_{rel}-\vec{v}$$</p> | |||
| <p>By the Pythagorean theorem</p> | |||
| <p class="exp">$$\fbox{${v}_\perp = 3{v}$}$$</p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\sin \alpha = u/v$$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> aliaksandr@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||