10. (10 points) As shown in the figure, there is a switch at point $A$ that can connect to $B$ or $C$. The circumference of the small loop track is 1.5 meters, and the circumference of the large loop track is 3 meters. Initially, $A$ is connected to $C$, and the train starts from point $A$, moving clockwise on the track. The switch changes the track connection every 1 minute. If the train's speed is 10 meters per minute, then the train will have used seconds when it returns to point $A$ for the 10th time.
【Analysis】To find out how much time is required, it is necessary to understand the equal relationship in this problem. Note that on the $A C$ track, if the switch suddenly changes to the $A B$ track, it will still go to point $A$ before taking the $A B$ track.

【Solution】According to the conditions, the small circle train takes $1.5 \div 10=0.15$ minutes to travel one lap, and the large circle train takes $3 \div 10=0.3$ minutes to travel one lap. Let the time taken to return to point $A$ be $t$ minutes. The following table can be obtained:
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline \begin{tabular}{c} 
Times of \\ 
returning to $A$ 
\end{tabular} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline
\end{tabular}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline \begin{tabular}{c} 
Time taken to \\ 
return to $A$ 
\end{tabular} & 0.3 & 0.6 & 0.9 & 1.2 & 1.35 & 1.5 & 1.65 & 1.8 & 1.95 & 2.1 \\
\hline \begin{tabular}{c} 
Track passed \\ 
through
\end{tabular} & $A C$ & $A C$ & $A C$ & $A B$ & $A B$ & $A B$ & $A B$ & $A B$ & $A B$ & $A C$ \\
\hline
\end{tabular}

Answer: The required time is 126 seconds.
Therefore, the answer is: 126.