<meta name="description" content="A website with solutions to physics problems from Savchenko Textbook">
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<meta name="description" content="The largest dataset of solutions of 'Savchenko. Problems in Physics'. Savchenko’s Problems in General Physics is widely used to prepare for olympiads and it is a useful tool to
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master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
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of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
$1.1.4.$ Three microphones located on the same straight line at points $A$, $B$, $C$ recorded successively at the moments $t_A > t_B > t_C$ the sound of an explosion that occurred at point $O$, which lies on the segment $AC$. Find the length of the segment $AO$ if $AB = BC = L$. At what point in time did the explosion occur?
<center>
<figure>
<img src="statement.png"
loading="lazy" width="230" />
<figcaption>
For problem 1.1.4
</figcaption>
</figure>
</center>
</p>
<h3>Solution</h3>
<p>
<p>
1. Let us introduce the following variables
</p>
<p>
$$x_B=L-x$$
</p>
<p>
$$t_A=t_B+\Delta t$$
</p>
<p>
2. Time of flight of $\gamma$-quantum to counters
</p>
<p>
$$t_B+\Delta t=\frac{x}{c}$$
</p>
<p>
$$t_B=\frac{L-x}{c}$$
</p>
<p>
3. Solving the equations together, we obtain
</p>
<p>
$$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$
</p>
<p>
$$x= \frac{L+c\Delta t}{2}=1.15\,м$$
</p>
</p>
<h4>Answer</h4>
<p>
At a distance $1.15\, \text{m}$ from the microphone $A$
<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>
<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="A website with solutions to physics problems from Savchenko Textbook">
<meta name="description" content="The largest dataset of solutions of 'Savchenko. Problems in Physics'. Savchenko’s Problems in General Physics is widely used to prepare for olympiads and it is a useful tool to
master and sharpen your skills and techniques in comptetitive problem solving. Some of these problems were a source
of inspiration for Jaan Kalda’s handouts and to some NBPhO problems. You may find problems from old IPhO
$1.1.4.$ Three microphones located on the same straight line at points $A$, $B$, $C$ recorded successively at the moments $t_A > t_B > t_C$ the sound of an explosion that occurred at point $O$, which lies on the segment $AC$. Find the length of the segment $AO$ if $AB = BC = L$. At what point in time did the explosion occur?
$1.1.4.$ Three microphones located on the same straight line at points $A$, $B$, $C$ recorded successively at the moments $t_A > t_B > t_C$ the sound of an explosion that occurred at point $O$, which lies on the segment $AC$. Find the length of the segment $AO$ if $AB = BC = L$. At what point in time did the explosion occur?
<center>
<center>
<figure>
<figure>
<img src="statement.png"
<img src="statement.png"
loading="lazy" width="230" />
loading="lazy" width="230" />
<figcaption>
<figcaption>
For problem 1.1.4
For problem 1.1.4
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
<p>
<p>
1. Let us introduce the following variables
1. Let us introduce the following variables
</p>
</p>
<p>
<p>
$$x_B=L-x$$
$$x_B=L-x$$
</p>
</p>
<p>
<p>
$$t_A=t_B+\Delta t$$
$$t_A=t_B+\Delta t$$
</p>
</p>
<p>
<p>
2. Time of flight of $\gamma$-quantum to counters
2. Time of flight of $\gamma$-quantum to counters
</p>
</p>
<p>
<p>
$$t_B+\Delta t=\frac{x}{c}$$
$$t_B+\Delta t=\frac{x}{c}$$
</p>
</p>
<p>
<p>
$$t_B=\frac{L-x}{c}$$
$$t_B=\frac{L-x}{c}$$
</p>
</p>
<p>
<p>
3. Solving the equations together, we obtain
3. Solving the equations together, we obtain
</p>
</p>
<p>
<p>
$$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$
$$\frac{L-x}{c} + \Delta t = \frac{x}{c}$$
</p>
</p>
<p>
<p>
$$x= \frac{L+c\Delta t}{2}=1.15\,м$$
$$x= \frac{L+c\Delta t}{2}=1.15\,м$$
</p>
</p>
</p>
</p>
<h4>Answer</h4>
<h4>Answer</h4>
<p>
<p>
At a distance $1.15\, \text{m}$ from the microphone $A$
At a distance $1.15\, \text{m}$ from the microphone $A$
<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>
<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>
