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| <meta property="og:description" content="A body for time t_0 moves with constant velocity v_0. Then its velocity increases linearly with time so that at time 2 t_0 it is equal to 2 v_0. Determine the path traveled by the body for time t > t_0."> | | <meta property="og:description" content="A body for time t_0 moves with constant velocity v_0. Then its velocity increases linearly with time so that at time 2 t_0 it is equal to 2 v_0. Determine the path traveled by the body for time t > t_0."> |
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| <title>A body for time t_0 moves with constant velocity v_0. Then its velocity increases linearly with time so that at time 2 t_0 it is equal to 2 v_0. Determine the path traveled by the body for time t > t_0.</title> | | <title>A body for time t_0 moves with constant velocity v_0. Then its velocity increases linearly with time so that at time 2 t_0 it is equal to 2 v_0. Determine the path traveled by the body for time t > t_0.</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.2.3.$ A body for time $t_0$ moves with constant velocity $v_0$. Then its velocity increases linearly with time so that at time $2 t_0$ it is equal to $2 v_0$. Determine the path traveled by the body for time $t > t_0$. | | $1.2.3.$ A body for time $t_0$ moves with constant velocity $v_0$. Then its velocity increases linearly with time so that at time $2 t_0$ it is equal to $2 v_0$. Determine the path traveled by the body for time $t > t_0$. |
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| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem $1.2.3$ | | For problem $1.2.3$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| | | |
| <p> | | <p> |
| At time $t_0$, the coordinate was $x_1=v_0t_0$. | | At time $t_0$, the coordinate was $x_1=v_0t_0$. |
| </p> | | </p> |
| <p> | | <p> |
| On the interval from $t_0$ to $2t_0$, the acceleration is constant and equal to $a=\frac{v_0}{t_0}$. | | On the interval from $t_0$ to $2t_0$, the acceleration is constant and equal to $a=\frac{v_0}{t_0}$. |
| </p> | | </p> |
| <p> | | <p> |
| In the case of equiaxcelerated motion with initial velocity $v_0$ from time $t_0$, the path is found as | | In the case of equiaxcelerated motion with initial velocity $v_0$ from time $t_0$, the path is found as |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$ | | $$ |
| x_2 = v_0 (t-t_0)+\frac{a(t-t_0)^2}{2} | | x_2 = v_0 (t-t_0)+\frac{a(t-t_0)^2}{2} |
| $$ | | $$ |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| | | |
| $$ | | $$ |
| x = x_1+x_2 | | x = x_1+x_2 |
| $$ | | $$ |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$ | | $$ |
| x = v_0 t+\frac{a(t-t_0)^2}{2} | | x = v_0 t+\frac{a(t-t_0)^2}{2} |
| $$ | | $$ |
| </p> | | </p> |
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| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$L = v_{0}t + \frac{v_{0} (t − t_{0})^{2}}{2t_{0}}$$ | | $$L = v_{0}t + \frac{v_{0} (t − t_{0})^{2}}{2t_{0}}$$ |
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