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streetmath [2020/03/08 11:13] – mario | streetmath [2020/03/08 16:19] – mario | ||
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Questions: | 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 $B$? | - 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 | + | - And what distance $D$ does the bicycle $B$ travel while the overtake takes place? |
=== Mathematical Model === | === Mathematical Model === | ||
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* $w$: | * $w$: | ||
+ | <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 $B$. | 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 $B$. | ||
+ | </ | ||
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 | 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 | ||
\[ T = \frac{w}{\Delta_v}.\] | \[ T = \frac{w}{\Delta_v}.\] | ||
- | Example: | + | <WRAP center round info 60%> |
- | It takes | + | Example: It takes |
\[ T | \[ T | ||
= \frac{10 \, | = \frac{10 \, | ||
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\] | \] | ||
for $C$ to overtake $B$ if their speed difference is 10 km/h. | for $C$ to overtake $B$ if their speed difference is 10 km/h. | ||
+ | </ | ||
Now, using $T$ and the reference speed $v_B$ of the cyclist, it is easy to calculate | Now, using $T$ and the reference speed $v_B$ of the cyclist, it is easy to calculate | ||
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\] | \] | ||
+ | <WRAP center round info 60%> | ||
Again, in our running example we have | Again, in our running example we have | ||
\[ | \[ | ||
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= 20 \,\text{m}. | = 20 \,\text{m}. | ||
\] | \] | ||
+ | </ | ||
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. | 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. | 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. | ||
+ | </ |