Differences

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--- streetmath [2020/03/08 12:00] – mario
+++ streetmath [2020/03/08 17:19] – mario
@@ Line 3: / Line 3: @@
 Questions:
   - Given the speed difference between $C$ and $B$ and the total speed of $B$, what time $T$ does it take for $C$ to overtake $B$?
-  - And what distance $D$ do both vehicles travel while the overtake takes place?
+  - What distance $D$ does the bicycle $B$ travel while the overtake takes place?
 === Mathematical Model ===
@@ Line 11: / Line 11: @@
   * $w$:   relative window size
+<WRAP center round info 60%>
 For example, the driver travels at $v_C$ = 30 km/h, the cyclist at $v_B$ = 20 km/h, and the relative window size is $w$ = 10 m, that is $C$ switches the lane 5 m behind $B$, passes $B$, and switches the lane again when $C$ is 5 meters in front of $B$.
+</WRAP>
 Consider the speed difference $\Delta_v = v_C - v_B$ of $C$ and $B$. Regarding question 1, take $B$ as the reference point. Then $C$ approaches $B$ with speed $\Delta_v$ and needs to travel a distance of $w$ in order to safely overtake $B$. We obtain
 \[ T = \frac{w}{\Delta_v}.\]
-Example:
+<WRAP center round info 60%>
+Example: It takes
 \[ T
 = \frac{10 \,\text{m}}{30 \frac{\,\text{km}}{\,\text{h}} - 20 \frac{\,\text{km}}{\,\text{h}}}
@@ Line 24: / Line 27: @@
 = 3.6 \,\text{s}
 \]
+for $C$ to overtake $B$ if their speed difference is 10 km/h.
+</WRAP>
+Now, using $T$ and the reference speed $v_B$ of the cyclist, it is easy to calculate
+\[
+D = v_B \cdot T.
+\]
+<WRAP center round info 60%>
+Again, in our running example we have
+\[
+D = 20 \,\frac{\text{km}}{\text{h}} \cdot 3.6 \,\text{s}
+= 20 \,\frac{1000\,\text{m}}{3600\,\text{s}} \cdot 3.6 \,\text{s}
+= 20 \,\text{m}.
+\]
+</WRAP>
+Now it remains to add the relative window size $w$ to $D$ in order to get the distance $C$ travels from the beginning of the overtake until its end.
+<WRAP center round info 60%>
+In total, overtaking a cyclist riding at 20 km/h while driving a car at 30 km/h takes 3.6 s and meanwhile the cyclist travels 20 m. In order to avoid hitting the cyclist, the driver needs to overview 30 m ahead.
+</WRAP>