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en/1.3.28.md
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| + | <meta name="description" content="Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points."> | ||
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| + | <meta property="og:description" content="Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points."> | ||
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| + | <title>Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points.</title> | ||
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| + | <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> | ||
| + | $1.3.28.$ Projectiles fly out at an initial velocity of $600$ $\frac{m}{s}$ at an angle of $30^\circ$, $45^\circ$, $60^\circ$ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points. | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.3.28/graph.png" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Graphs of $y(x)$ for three angles | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p>At the initial moment of time, the normal component of $\vec{g}$, perpendicular to $\vec{v}$:</p> | ||
| + | <p class="exp"> | ||
| + | $$g_\perp = g \cos \alpha$$ | ||
| + | </p> | ||
| + | <p>Then the radius of curvature at the initial point is found as </p> | ||
| + | <p class="exp"> | ||
| + | $$ \fbox{$R = \frac{v^2}{g \cos \alpha}$} $$ | ||
| + | </p> | ||
| + | <p>At the top point of the trajectory, the acceleration will coincide with the acceleration of gravity, and the speed will be equal to the horizontal</p> | ||
| + | <p class="exp"> | ||
| + | $$g_\perp = g$$ | ||
| + | $$v = v_0 \cos \alpha$$ | ||
| + | </p> | ||
| + | <p>Similarly, the radius of curvature at this point is found as </p> | ||
| + | <p class="exp"> | ||
| + | $$ \fbox{$R = \frac{v^2 \cos^2 \alpha}{g}$} $$ | ||
| + | </p> | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$27.5\text{ and }42.4\text{ km; }18.3\text{ and }52\text{ km; }0.2\text{ and }73.4\text{ km}$$ | ||
| + | </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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| @@ -0,0 +1,99 @@ | |||
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| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
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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="Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points."> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points."> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>Projectiles fly out at an initial velocity of 600 \frac{m}{s} at an angle of 30^\circ, 45^\circ, 60^\circ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points.</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="../../#1.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.3.28.$ Projectiles fly out at an initial velocity of $600$ $\frac{m}{s}$ at an angle of $30^\circ$, $45^\circ$, $60^\circ$ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points. | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.3.28/graph.png" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Graphs of $y(x)$ for three angles | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p>At the initial moment of time, the normal component of $\vec{g}$, perpendicular to $\vec{v}$:</p> | |||
| <p class="exp"> | |||
| $$g_\perp = g \cos \alpha$$ | |||
| </p> | |||
| <p>Then the radius of curvature at the initial point is found as </p> | |||
| <p class="exp"> | |||
| $$ \fbox{$R = \frac{v^2}{g \cos \alpha}$} $$ | |||
| </p> | |||
| <p>At the top point of the trajectory, the acceleration will coincide with the acceleration of gravity, and the speed will be equal to the horizontal</p> | |||
| <p class="exp"> | |||
| $$g_\perp = g$$ | |||
| $$v = v_0 \cos \alpha$$ | |||
| </p> | |||
| <p>Similarly, the radius of curvature at this point is found as </p> | |||
| <p class="exp"> | |||
| $$ \fbox{$R = \frac{v^2 \cos^2 \alpha}{g}$} $$ | |||
| </p> | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$27.5\text{ and }42.4\text{ km; }18.3\text{ and }52\text{ km; }0.2\text{ and }73.4\text{ km}$$ | |||
| </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> | |||