Edit to “Statement”
en/1.1.4.md
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| @@ -1,6 +1,7 @@ | |||
| ### Statement | |||
| − | $1.1.4.$ 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? | ||
| + | $1.1.4.$ Counters \(A\) and \(B\), which register the moment of the arrival of a \(\gamma\)-quantum, are located at a distance of \(2\,\mathrm{m}\) from each other. At a point between them a \(\pi_0\) meson decayed into two \(\gamma\)-quanta. Find the position of this point if counter \(A\) detected the \(\gamma\)-quantum \(10^{-9}\,\mathrm{s}\) later than counter \(B\). The speed of light is \(3\times10^8\,\mathrm{m/s}\). | ||
| + | |||
|  | |||
| ### Solution | |||
| 1\. Let us introduce the following variables | |||
| $$ | |||
| x_B=L-x | |||
| $$ | |||
| $$ | |||
| t_A=t_B+\Delta t | |||
| $$ | |||
| 2\. Time of flight of $\gamma$-quantum to counters | |||
| $$ | |||
| t_B+\Delta t=\frac{x}{c} | |||
| $$ | |||
| $$ | |||
| t_B=\frac{L-x}{c} | |||
| $$ | |||
| 3\. Solving the equations together, we obtain | |||
| $$ | |||
| \frac{L-x}{c} + \Delta t = \frac{x}{c} | |||
| $$ | |||
| $$ | |||
| x= \frac{L+c\Delta t}{2}=1.15\text{ m} | |||
| $$ | |||
| #### Answer | |||
| At a distance $1.15\text{ m}$ from the microphone $A$ | |||
| unchanged lines 43 | |||
| @@ -1,6 +1,7 @@ | |||
| ### Statement | ### Statement | ||
| $1.1.4.$ 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? | $1.1.4.$ Counters \(A\) and \(B\), which register the moment of the arrival of a \(\gamma\)-quantum, are located at a distance of \(2\,\mathrm{m}\) from each other. At a point between them a \(\pi_0\) meson decayed into two \(\gamma\)-quanta. Find the position of this point if counter \(A\) detected the \(\gamma\)-quantum \(10^{-9}\,\mathrm{s}\) later than counter \(B\). The speed of light is \(3\times10^8\,\mathrm{m/s}\). | ||
|  |  | ||
| ### Solution | ### Solution | ||
| 1\. Let us introduce the following variables | 1\. Let us introduce the following variables | ||
| $$ | $$ | ||
| x_B=L-x | x_B=L-x | ||
| $$ | $$ | ||
| $$ | $$ | ||
| t_A=t_B+\Delta t | t_A=t_B+\Delta t | ||
| $$ | $$ | ||
| 2\. Time of flight of $\gamma$-quantum to counters | 2\. Time of flight of $\gamma$-quantum to counters | ||
| $$ | $$ | ||
| t_B+\Delta t=\frac{x}{c} | t_B+\Delta t=\frac{x}{c} | ||
| $$ | $$ | ||
| $$ | $$ | ||
| t_B=\frac{L-x}{c} | t_B=\frac{L-x}{c} | ||
| $$ | $$ | ||
| 3\. Solving the equations together, we obtain | 3\. Solving the equations together, we obtain | ||
| $$ | $$ | ||
| \frac{L-x}{c} + \Delta t = \frac{x}{c} | \frac{L-x}{c} + \Delta t = \frac{x}{c} | ||
| $$ | $$ | ||
| $$ | $$ | ||
| x= \frac{L+c\Delta t}{2}=1.15\text{ m} | x= \frac{L+c\Delta t}{2}=1.15\text{ m} | ||
| $$ | $$ | ||
| #### Answer | #### Answer | ||
| At a distance $1.15\text{ m}$ from the microphone $A$ | At a distance $1.15\text{ m}$ from the microphone $A$ | ||
| unchanged lines 43 | |||