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en/1.3.24.md
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| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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| + | <title>At a time when the particle velocity is 10^6 \frac{m}{s}, its acceleration is 10^4 \frac{m}{s^2} and is directed at an angle of 30^\circ to the velocity. How much will the speed increase in 10^{-2} seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment?</title> | ||
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| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.3.24.$ At a time when the particle velocity is $10^6$ $\frac{m}{s}$, its acceleration is $10^4$ $\frac{m}{s^2}$ and is directed at an angle of $30^\circ$ to the velocity. How much will the speed increase in $10^{-2}$ seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment? | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <p>We decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$:</p> | ||
| + | <p class="exp"> | ||
| + | $$ a_\parallel = a \cdot \cos 30^{\circ} = a \frac{\sqrt{3}}{2} $$ | ||
| + | </p> | ||
| + | <p class="exp"> | ||
| + | $$ a_\perp = a \cdot \sin 30^{\circ}= a \frac{1}{2} $$ | ||
| + | </p> | ||
| + | <p>During time $\Delta t$ the speed will increase by $\Delta v_\perp = a_\perp \Delta t$ and $\Delta v _\parallel = a_\parallel \Delta t$, in the transverse and longitudinal directions, respectively.</p> | ||
| + | <p>New speed values:</p> | ||
| + | <p class="exp"> | ||
| + | $$ v_\parallel = \frac{\sqrt{3}}{2}a \Delta t +v_0 $$ | ||
| + | </p> | ||
| + | <p class="exp"> | ||
| + | $$ v_\perp = \frac{1}{2}a \Delta t $$ | ||
| + | </p> | ||
| + | <p>Thus, the desired angle:</p> | ||
| + | <p class="exp"> | ||
| + | $$ \tan \alpha = \frac{v_\perp}{v_\parallel} $$ | ||
| + | </p> | ||
| + | |||
| + | <p>Angular velocity</p> | ||
| + | |||
| + | <p class="exp"> | ||
| + | $$ \omega = \frac{\alpha}{t} $$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $\text{With}(\sqrt{3}/2)\cdot10^2\text{ m/s;}$ на $5\cdot10^{-5}\text{ rad;}$ $\omega=5\cdot10^{-3}\text{ s}^{-1}.$ | ||
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| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
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| + | <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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| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="At a time when the particle velocity is 10^6 \frac{m}{s}, its acceleration is 10^4 \frac{m}{s^2} and is directed at an angle of 30^\circ to the velocity. How much will the speed increase in 10^{-2} seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="At a time when the particle velocity is 10^6 \frac{m}{s}, its acceleration is 10^4 \frac{m}{s^2} and is directed at an angle of 30^\circ to the velocity. How much will the speed increase in 10^{-2} seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment?"> | |||
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| </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="../../#1.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.3.24.$ At a time when the particle velocity is $10^6$ $\frac{m}{s}$, its acceleration is $10^4$ $\frac{m}{s^2}$ and is directed at an angle of $30^\circ$ to the velocity. How much will the speed increase in $10^{-2}$ seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment? | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <p>We decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$:</p> | |||
| <p class="exp"> | |||
| $$ a_\parallel = a \cdot \cos 30^{\circ} = a \frac{\sqrt{3}}{2} $$ | |||
| </p> | |||
| <p class="exp"> | |||
| $$ a_\perp = a \cdot \sin 30^{\circ}= a \frac{1}{2} $$ | |||
| </p> | |||
| <p>During time $\Delta t$ the speed will increase by $\Delta v_\perp = a_\perp \Delta t$ and $\Delta v _\parallel = a_\parallel \Delta t$, in the transverse and longitudinal directions, respectively.</p> | |||
| <p>New speed values:</p> | |||
| <p class="exp"> | |||
| $$ v_\parallel = \frac{\sqrt{3}}{2}a \Delta t +v_0 $$ | |||
| </p> | |||
| <p class="exp"> | |||
| $$ v_\perp = \frac{1}{2}a \Delta t $$ | |||
| </p> | |||
| <p>Thus, the desired angle:</p> | |||
| <p class="exp"> | |||
| $$ \tan \alpha = \frac{v_\perp}{v_\parallel} $$ | |||
| </p> | |||
| <p>Angular velocity</p> | |||
| <p class="exp"> | |||
| $$ \omega = \frac{\alpha}{t} $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $\text{With}(\sqrt{3}/2)\cdot10^2\text{ m/s;}$ на $5\cdot10^{-5}\text{ rad;}$ $\omega=5\cdot10^{-3}\text{ s}^{-1}.$ | |||
| </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> | |||
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| </html> | |||