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| + | <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | ||
| + | <meta name="description" content="A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus."> | ||
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| + | <meta property="og:description" content="A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus."> | ||
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| + | <title>A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus.</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> | ||
| + | $1.4.14.$ A nucleus traveling at speed $v$ splits into two identical fragments. Determine the maximum possible angle $\alpha$ between the velocities of one of the fragments and the vector $v$, if the fragments have a velocity $u < v$ during the decay of a resting nucleus. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <p>The velocity and one of the accelerations of the nucleus are</p> | ||
| + | <p class="exp">$$ \vec{u} = \vec{v}_{0} + \vec{v} $$</p> | ||
| + | |||
| + | <p>where $ \vec{v}_{0} $ is its velocity before the decay of the nucleus, $ \vec{v} $ is the velocity of the fragment relative to the resting nucleus (that is, in the coordinate system in which the nucleus was at rest before the decay).</p> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.4.14/draw.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | Vector image | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <p>Let's arrange the vector $ \vec{v}_{0} $ as shown in the figure. The end of the vector can lie on a circle of radius $| \vec{v} |$. </p> | ||
| + | <p>It is clear from the figure that the vector and forms the maximum angle $ \alpha $ with the vector $ \vec{v}_{0} $ if the angle between the vectors $ \vec{u} $ and $ \vec{v} $ is $ \pi / 2 $. In this case</p> | ||
| + | |||
| + | <p class="exp">$$\fbox{$ \alpha = \arcsin \frac{v}{v_0} $}$$</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> | ||
| @@ -0,0 +1,92 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <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="A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>A nucleus traveling at speed v splits into two identical fragments. Determine the maximum possible angle \alpha between the velocities of one of the fragments and the vector v, if the fragments have a velocity u < v during the decay of a resting nucleus.</title> | |||
| <link rel="stylesheet" href="https://savchenkosolutions.com/css/css-latex/style.css"> | |||
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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> | |||
| $1.4.14.$ A nucleus traveling at speed $v$ splits into two identical fragments. Determine the maximum possible angle $\alpha$ between the velocities of one of the fragments and the vector $v$, if the fragments have a velocity $u < v$ during the decay of a resting nucleus. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <p>The velocity and one of the accelerations of the nucleus are</p> | |||
| <p class="exp">$$ \vec{u} = \vec{v}_{0} + \vec{v} $$</p> | |||
| <p>where $ \vec{v}_{0} $ is its velocity before the decay of the nucleus, $ \vec{v} $ is the velocity of the fragment relative to the resting nucleus (that is, in the coordinate system in which the nucleus was at rest before the decay).</p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.4.14/draw.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| Vector image | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p>Let's arrange the vector $ \vec{v}_{0} $ as shown in the figure. The end of the vector can lie on a circle of radius $| \vec{v} |$. </p> | |||
| <p>It is clear from the figure that the vector and forms the maximum angle $ \alpha $ with the vector $ \vec{v}_{0} $ if the angle between the vectors $ \vec{u} $ and $ \vec{v} $ is $ \pi / 2 $. In this case</p> | |||
| <p class="exp">$$\fbox{$ \alpha = \arcsin \frac{v}{v_0} $}$$</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> | |||