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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.">
<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>
$1.3.28.$ Projectiles fly out at an initial velocity of $600$$\frac{m}{s}$ at an angle of $30^\circ$, $45^\circ$, $60^\circ$ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points.
<p>At the initial moment of time, the normal component of $\vec{g}$, perpendicular to $\vec{v}$:</p>
<p class="exp">
$$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}$$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
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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="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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@@ -14,9 +14,9 @@
<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 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.">
<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>
<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>
$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.
$1.3.28.$ Projectiles fly out at an initial velocity of $600$$\frac{m}{s}$ at an angle of $30^\circ$, $45^\circ$, $60^\circ$ to the horizon. Determine the radius of curvature of the projectile trajectory at their highest and starting points.
<p>At the initial moment of time, the normal component of $\vec{g}$, perpendicular to $\vec{v}$:</p>
<p>At the initial moment of time, the normal component of $\vec{g}$, perpendicular to $\vec{v}$:</p>
<p class="exp">
<p class="exp">
$$g_\perp = g \cos\alpha$$
$$g_\perp = g \cos\alpha$$
</p>
</p>
<p>Then the radius of curvature at the initial point is found as </p>
<p>Then the radius of curvature at the initial point is found as </p>
<p class="exp">
<p class="exp">
$$\fbox{$R = \frac{v^2}{g \cos\alpha}$}$$
$$\fbox{$R = \frac{v^2}{g \cos\alpha}$}$$
</p>
</p>
<p>At the top point of the trajectory, the acceleration will coincide with the acceleration of gravity, and the speed will be equal to the horizontal</p>
<p>At the top point of the trajectory, the acceleration will coincide with the acceleration of gravity, and the speed will be equal to the horizontal</p>
<p class="exp">
<p class="exp">
$$g_\perp = g$$
$$g_\perp = g$$
$$v = v_0 \cos\alpha$$
$$v = v_0 \cos\alpha$$
</p>
</p>
<p>Similarly, the radius of curvature at this point is found as </p>
<p>Similarly, the radius of curvature at this point is found as </p>
<p class="exp">
<p class="exp">
$$\fbox{$R = \frac{v^2 \cos^2 \alpha}{g}$}$$
$$\fbox{$R = \frac{v^2 \cos^2 \alpha}{g}$}$$
</p>
</p>
</p>
</p>
<h4>Answer</h4>
<h4>Answer</h4>
<p>
<p>
$$27.5\text{ and }42.4\text{ km; }18.3\text{ and }52\text{ km; }0.2\text{ and }73.4\text{ km}$$
$$27.5\text{ and }42.4\text{ km; }18.3\text{ and }52\text{ km; }0.2\text{ and }73.4\text{ km}$$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
