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| + | <meta name="description" content="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
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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="../#1.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.2.14.$ According to the graph of acceleration versus time, set the speed at times $4$ and $15$ s, if at time $1$ s the speed is $3$ $\frac{m}{s}$. | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="statement.png" | ||
| + | loading="lazy" width="230" /> | ||
| + | <figcaption> | ||
| + | For problem $1.2.14$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | Acceleration depends on time, as: | ||
| + | $$\left\{\begin{matrix} | ||
| + | a(t) = 0\;m/s^2,\; 0\text{ s} \leq t \leq 2\text{ s}\\ | ||
| + | a(t) = 20t,\; 2\text{ s} \leq t \leq 5\text{ s}\\ | ||
| + | a(t) = 60\;m/s^2,\; 5\text{ s} \leq t \leq 9\text{ s}\\ | ||
| + | a(t) = 60-20(t-9),\; 9\text{ s} \leq t \leq 12\text{ s}\\ | ||
| + | a(t) = 0, t> 12\text{ s} | ||
| + | \end{matrix}\right.$$ | ||
| + | |||
| + | Given that the area under the graph of acceleration vs. time is velocity vs. time. Then the velocity depends on time as: | ||
| + | |||
| + | $$\left\{\begin{matrix} | ||
| + | v(t) = 3\;m/s,\; 0\text{ s} \leq t \leq 2\text{ s}\\ | ||
| + | v(t) = 3+20t^2/2,\; 2\text{ s} \leq t \leq 5\text{ s}\\ | ||
| + | v(t) = 93+60(t-5),\; 5\text{ s} \leq t \leq 9\text{ s}\\ | ||
| + | v(t) = 333 - 20 (t - 9)^2/2 + 60 (t - 9),\; 9\text{ s} \leq t \leq 12\text{ s}\\ | ||
| + | v(t) = 423\;m/s, t> 12\text{ s} | ||
| + | \end{matrix}\right.$$ | ||
| + | |||
| + | The graph of this dependence is presented below: | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="graph.png" alt="1.2.14" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Dependence of velocity on time | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | It follows that at $4$ and $15\text{ s}$ the velocity is $43$ and $423\;m/s$, respectively. | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$v_1=43\;m/s; \;v_2=423\;m/s.$$ | ||
| + | </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> astrosander01@gmail.com <br></small> | ||
| + | </p> | ||
| + | </footer> | ||
| + | </body> | ||
| + | |||
| + | </html> | ||
| @@ -0,0 +1,117 @@ | |||
| <!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="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}."> | |||
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| <meta property="og:description" content="According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}."> | |||
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| <title>According to the graph of acceleration versus time, set the speed at times 4 and 15 s, if at time 1 s the speed is 3 \frac{m}{s}.</title> | |||
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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="../#1.2">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.2.14.$ According to the graph of acceleration versus time, set the speed at times $4$ and $15$ s, if at time $1$ s the speed is $3$ $\frac{m}{s}$. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="statement.png" | |||
| loading="lazy" width="230" /> | |||
| <figcaption> | |||
| For problem $1.2.14$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| Acceleration depends on time, as: | |||
| $$\left\{\begin{matrix} | |||
| a(t) = 0\;m/s^2,\; 0\text{ s} \leq t \leq 2\text{ s}\\ | |||
| a(t) = 20t,\; 2\text{ s} \leq t \leq 5\text{ s}\\ | |||
| a(t) = 60\;m/s^2,\; 5\text{ s} \leq t \leq 9\text{ s}\\ | |||
| a(t) = 60-20(t-9),\; 9\text{ s} \leq t \leq 12\text{ s}\\ | |||
| a(t) = 0, t> 12\text{ s} | |||
| \end{matrix}\right.$$ | |||
| Given that the area under the graph of acceleration vs. time is velocity vs. time. Then the velocity depends on time as: | |||
| $$\left\{\begin{matrix} | |||
| v(t) = 3\;m/s,\; 0\text{ s} \leq t \leq 2\text{ s}\\ | |||
| v(t) = 3+20t^2/2,\; 2\text{ s} \leq t \leq 5\text{ s}\\ | |||
| v(t) = 93+60(t-5),\; 5\text{ s} \leq t \leq 9\text{ s}\\ | |||
| v(t) = 333 - 20 (t - 9)^2/2 + 60 (t - 9),\; 9\text{ s} \leq t \leq 12\text{ s}\\ | |||
| v(t) = 423\;m/s, t> 12\text{ s} | |||
| \end{matrix}\right.$$ | |||
| The graph of this dependence is presented below: | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="graph.png" alt="1.2.14" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Dependence of velocity on time | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| It follows that at $4$ and $15\text{ s}$ the velocity is $43$ and $423\;m/s$, respectively. | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$v_1=43\;m/s; \;v_2=423\;m/s.$$ | |||
| </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> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||