<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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?">
<meta property="og:title" content="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?">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="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?">
<title>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?</title>
$1.3.24.$ 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>
<h3>Solution</h3>
<p>
<p>We decompose the acceleration into two components, tangential $a_\parallel$ and normal $a_\perp$:</p>
<p class="exp">
$$ a_\parallel = a \cdot\cos 30^{\circ} = a \frac{\sqrt{3}}{2}$$
</p>
<p class="exp">
$$ a_\perp = a \cdot\sin 30^{\circ}= a \frac{1}{2}$$
</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 class="exp">
$$ v_\parallel = \frac{\sqrt{3}}{2}a \Delta t +v_0 $$
</p>
<p class="exp">
$$ v_\perp = \frac{1}{2}a \Delta t $$
</p>
<p>Thus, the desired angle:</p>
<p class="exp">
$$\tan\alpha = \frac{v_\perp}{v_\parallel}$$
</p>
<p>Angular velocity</p>
<p class="exp">
$$\omega = \frac{\alpha}{t}$$
</p>
</p>
<h4>Answer</h4>
<p>
$\text{With}(\sqrt{3}/2)\cdot10^2\text{ m/s;}$ на $5\cdot10^{-5}\text{ rad;}$$\omega=5\cdot10^{-3}\text{ s}^{-1}.$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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?">
<meta name="description" content="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?">
<meta property="og:title" content="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?">
<meta property="og:title" content="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?">
<meta property="og:image" content="img/logo.png">
<meta property="og:image" content="img/logo.png">
@@ -14,9 +14,9 @@
<meta property="og:description" content="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?">
<meta property="og:description" content="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?">
<title>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?</title>
<title>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?</title>
$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 $$
</p>
</p>
<p>Thus, the desired angle:</p>
<p>Thus, the desired angle:</p>
<p class="exp">
<p class="exp">
$$\tan\alpha = \frac{v_\perp}{v_\parallel}$$
$$\tan\alpha = \frac{v_\perp}{v_\parallel}$$
</p>
</p>
<p>Angular velocity</p>
<p>Angular velocity</p>
<p class="exp">
<p class="exp">
$$\omega = \frac{\alpha}{t}$$
$$\omega = \frac{\alpha}{t}$$
</p>
</p>
</p>
</p>
<h4>Answer</h4>
<h4>Answer</h4>
<p>
<p>
$\text{With}(\sqrt{3}/2)\cdot10^2\text{ m/s;}$ на $5\cdot10^{-5}\text{ rad;}$$\omega=5\cdot10^{-3}\text{ s}^{-1}.$
$\text{With}(\sqrt{3}/2)\cdot10^2\text{ m/s;}$ на $5\cdot10^{-5}\text{ rad;}$$\omega=5\cdot10^{-3}\text{ s}^{-1}.$
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
<small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small>
