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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> | | <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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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
| </p> | | </p> |
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| <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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. | | $1.3.28.$ 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> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <img src="https://savchenkosolutions.com/1/1.3.28/graph.png" | | <img src="https://savchenkosolutions.com/1/1.3.28/graph.png" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Graphs of $y(x)$ for three angles | | Graphs of $y(x)$ for three angles |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <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"> |
| $$g_\perp = g$$ | | $$g_\perp = g$$ |
| $$v = v_0 \cos \alpha$$ | | $$v = v_0 \cos \alpha$$ |
