<meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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? For problem 1.1.5">
<meta property="og:title" content="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? For problem 1.1.5">
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@@ -14,9 +14,9 @@
<meta property="og:description" content="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? For problem 1.1.5">
<title>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? For problem 1.1.5</title>
$1.1.5^*.$ 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 1.1.5
</figcaption>
</figure>
</center>
</p>
<h3>Solution</h3>
<p>
<p class="TxtSolutions">
Let the explosion occurred at time $t_0$, then the time of signal registration at point $A$ is equal to:<br>
</p>
$$t_A = t_0 + \frac{L + x}{c}, \, (1)$$
<p class="TxtSolutions">
where $x = BO$.<br><br>
Similar to point $B$<br>
</p>
<p style="text-align: center;">
$$t_B = t_0 + \frac{x}{c}, \,(2)$$
</p>
<p class="TxtSolutions">
For point $C$<br>
</p>
<p style="text-align: center;">
$$t_C = t_0 + \frac{L - x}{c}, \,(3)$$
</p>
<p class="TxtSolutions">
Let's subtract the second equation from the first equation:<br>
</p>
<p style="text-align: center;">
$$t_A – t_B = \frac{L}{c}. \,(4)$$
</p>
<p class="TxtSolutions">
And from the first equation we subtract the third equation<br>
</p>
<p style="text-align: center;">
$$t_A – t_C = \frac{2x}{c}. \,(5)$$
</p>
<p class="TxtSolutions">
From equation $(4)$ let's express $c = \frac{L}{t_A – t_B}$, а из $(5)$$x = \frac{t_A – t_C}{2}\cdot c$. Then the required distance<br>
</p>
<p style="text-align: center;">
$$AO = L + x = L + \frac{t_A – t_C}{2}\frac{L}{t_A – t_B}$$
<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="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda">
<meta name="description" content="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? For problem 1.1.5">
<meta name="description" content="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? For problem 1.1.5">
<meta property="og:title" content="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? For problem 1.1.5">
<meta property="og:title" content="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? For problem 1.1.5">
<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="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? For problem 1.1.5">
<meta property="og:description" content="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? For problem 1.1.5">
<title>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? For problem 1.1.5</title>
<title>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? For problem 1.1.5</title>
$1.1.5^*.$ 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?
$1.1.5^*.$ 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 1.1.5
For problem 1.1.5
</figcaption>
</figcaption>
</figure>
</figure>
</center>
</center>
</p>
</p>
<h3>Solution</h3>
<h3>Solution</h3>
<p>
<p>
<p class="TxtSolutions">
<p class="TxtSolutions">
Let the explosion occurred at time $t_0$, then the time of signal registration at point $A$ is equal to:<br>
Let the explosion occurred at time $t_0$, then the time of signal registration at point $A$ is equal to:<br>
</p>
</p>
$$t_A = t_0 + \frac{L + x}{c}, \, (1)$$
$$t_A = t_0 + \frac{L + x}{c}, \, (1)$$
<p class="TxtSolutions">
<p class="TxtSolutions">
where $x = BO$.<br><br>
where $x = BO$.<br><br>
Similar to point $B$<br>
Similar to point $B$<br>
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$$t_B = t_0 + \frac{x}{c}, \,(2)$$
$$t_B = t_0 + \frac{x}{c}, \,(2)$$
</p>
</p>
<p class="TxtSolutions">
<p class="TxtSolutions">
For point $C$<br>
For point $C$<br>
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$$t_C = t_0 + \frac{L - x}{c}, \,(3)$$
$$t_C = t_0 + \frac{L - x}{c}, \,(3)$$
</p>
</p>
<p class="TxtSolutions">
<p class="TxtSolutions">
Let's subtract the second equation from the first equation:<br>
Let's subtract the second equation from the first equation:<br>
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$$t_A – t_B = \frac{L}{c}. \,(4)$$
$$t_A – t_B = \frac{L}{c}. \,(4)$$
</p>
</p>
<p class="TxtSolutions">
<p class="TxtSolutions">
And from the first equation we subtract the third equation<br>
And from the first equation we subtract the third equation<br>
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$$t_A – t_C = \frac{2x}{c}. \,(5)$$
$$t_A – t_C = \frac{2x}{c}. \,(5)$$
</p>
</p>
<p class="TxtSolutions">
<p class="TxtSolutions">
From equation $(4)$ let's express $c = \frac{L}{t_A – t_B}$, а из $(5)$$x = \frac{t_A – t_C}{2}\cdot c$. Then the required distance<br>
From equation $(4)$ let's express $c = \frac{L}{t_A – t_B}$, а из $(5)$$x = \frac{t_A – t_C}{2}\cdot c$. Then the required distance<br>
</p>
</p>
<p style="text-align: center;">
<p style="text-align: center;">
$$AO = L + x = L + \frac{t_A – t_C}{2}\frac{L}{t_A – t_B}$$
$$AO = L + x = L + \frac{t_A – t_C}{2}\frac{L}{t_A – t_B}$$
<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>
