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| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
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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> | | <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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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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 $3$ $\frac{m}{s}$. | | $1.2.14.$ 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 $3$ $\frac{m}{s}$. |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem $1.2.14$ | | For problem $1.2.14$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| Acceleration depends on time, as: | | Acceleration depends on time, as: |
| $$\left\{\begin{matrix} | | $$\left\{\begin{matrix} |
| a(t) = 0\;m/s^2,\; 0\text{ s} \leq t \leq 2\text{ s}\\ | | 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) = 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\;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) = 60-20(t-9),\; 9\text{ s} \leq t \leq 12\text{ s}\\ |
| a(t) = 0, t> 12\text{ s} | | a(t) = 0, t> 12\text{ s} |
| \end{matrix}\right.$$ | | \end{matrix}\right.$$ |
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| Given that the area under the graph of acceleration vs. time is velocity vs. time. Then the velocity depends on time as: | | Given that the area under the graph of acceleration vs. time is velocity vs. time. Then the velocity depends on time as: |
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| $$\left\{\begin{matrix} | | $$\left\{\begin{matrix} |
| v(t) = 3\;m/s,\; 0\text{ s} \leq t \leq 2\text{ s}\\ | | 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) = 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) = 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) = 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} | | v(t) = 423\;m/s, t> 12\text{ s} |
| \end{matrix}\right.$$ | | \end{matrix}\right.$$ |
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| The graph of this dependence is presented below: | | The graph of this dependence is presented below: |
| </p> | | </p> |
| <center> | | <center> |
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| <img src="graph.png" alt="1.2.14" | | <img src="graph.png" alt="1.2.14" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Dependence of velocity on time | | Dependence of velocity on time |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| It follows that at $4$ and $15\text{ s}$ the velocity is $43$ and $423\;m/s$, respectively. | | It follows that at $4$ and $15\text{ s}$ the velocity is $43$ and $423\;m/s$, respectively. |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$v_1=43\;m/s; \;v_2=423\;m/s.$$ | | $$v_1=43\;m/s; \;v_2=423\;m/s.$$ |
| </p> | | </p> |
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