<style>
.room-tag {
background-color: #ffffff;
color: #000000;
display: inline-block;
padding: 1px 5px;
border-radius: 8px;
text-align:justify;
font-size:22px;
font-weight:600;
}
</style>

# CHAPTER 2 Problem 07

<span class="room-tag">Two trains, each having a speed of 30 km/h, are headed at each other on the same straight track. A bird that can fly 60 km/h flies off the front of one train when they are 60 km apart and heads directly for the other train. On reaching the other train, the (crazy) bird flies directly back to the first train, and so forth. What is the total distance the bird travels before the trains collide?</span>

<br/>

第一次從A車頭飛到B車頭的距離：$L_1=60(\text{km})$<br>
鳥與車頭相向而行，飛行時間為
\begin{aligned}
\Delta t_1={60 \over (30+60)}={2 \over 3}(\text{H})
\end{aligned}

<br>
<br>

第二次從A車頭到B車頭的距離：
\begin{aligned}
L_2 =60-60\times {2 \over 3}=20(\text{km})
\end{aligned}

鳥與車頭相向而行，飛行時間為
\begin{aligned}
\Delta t_2={20 \over (30+60)}={2 \over 9}(\text{H})
\end{aligned}

<br>
<br>

第三次從A車頭到B車頭的距離：
\begin{aligned}
L_3 =20-60\times {2 \over 9}={20 \over 3}(\text{km})
\end{aligned}

鳥與車頭相向而行，飛行時間為
\begin{aligned}
\Delta t_3={{20 \over 3}\over (30+60)}={2 \over 27}(\text{H})
\end{aligned}

<br>
<br>

歸納整理出$\Delta t$之關係式為
\begin{aligned}
\Delta t_{n}=2\times {1 \over 3^n}
\end{aligned}

鳥總飛行時間為
\begin{aligned}
\sum_{i=1}^{\infty} \Delta t_i={{2\over 3} \over {1-{1 \over 3}}}=1(\text{H})
\end{aligned}

總飛行距離為
\begin{aligned}
\ell=60\times 1=60(\text{km})
\end{aligned}

###### tag: `Fundamentals of Physics`