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| <div id="logo"> | | <div id="logo"> |
| <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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| </a> | | </a> |
| <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> |
| $1.3.24.$ At a time when the particle velocity is $10^6$ $\frac{m}{s}$, its acceleration is $10^4$ $\frac{m}{s^2}$ and is directed at an angle of $30^\circ$ to the velocity. How much will the speed increase in $10^{-2}$ seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment? | | $1.3.24.$ At a time when the particle velocity is $10^6$ $\frac{m}{s}$, its acceleration is $10^4$ $\frac{m}{s^2}$ and is directed at an angle of $30^\circ$ to the velocity. How much will the speed increase in $10^{-2}$ seconds? At what angle will the speed direction change? What is the angular velocity of rotation of the velocity vector at this moment? |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <p>We decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$:</p> | | <p>We decompose the acceleration into two components, tangential $a_\parallel $ and normal $a_\perp$:</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ a_\parallel = a \cdot \cos 30^{\circ} = a \frac{\sqrt{3}}{2} $$ | | $$ a_\parallel = a \cdot \cos 30^{\circ} = a \frac{\sqrt{3}}{2} $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ a_\perp = a \cdot \sin 30^{\circ}= a \frac{1}{2} $$ | | $$ a_\perp = a \cdot \sin 30^{\circ}= a \frac{1}{2} $$ |
| </p> | | </p> |
| <p>During time $\Delta t$ the speed will increase by $\Delta v_\perp = a_\perp \Delta t$ and $\Delta v _\parallel = a_\parallel \Delta t$, in the transverse and longitudinal directions, respectively.</p> | | <p>During time $\Delta t$ the speed will increase by $\Delta v_\perp = a_\perp \Delta t$ and $\Delta v _\parallel = a_\parallel \Delta t$, in the transverse and longitudinal directions, respectively.</p> |
| <p>New speed values:</p> | | <p>New speed values:</p> |
| <p class="exp"> | | <p class="exp"> |
| $$ v_\parallel = \frac{\sqrt{3}}{2}a \Delta t +v_0 $$ | | $$ v_\parallel = \frac{\sqrt{3}}{2}a \Delta t +v_0 $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ v_\perp = \frac{1}{2}a \Delta t $$ | | $$ v_\perp = \frac{1}{2}a \Delta t $$ |
