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Consider the following problem.
A train has to follow a track. The speed of the train $S_f$ is $1$ km/hr if the train is traveling on the a straight track, and $v$ km/hr if the train is traveling over a $45^\circ$ curve. What is the minimum speed at which the train can travel without collision in the straight track?
We know that the formula for speed (as shown in the image) has $\tan{(\frac{\pi}{2} -\theta)}\times$ in the numerator and $\tan{(\frac{\pi}{2} +\theta)}\times$ in the denominator.
If $S_f$ is greater than $1$, i.e., if $v$ is greater than $1$, i.e., if there are more radians in the numerator than in the denominator, then it is obvious that the train has to travel at a lesser speed, and hence it will avoid collision.
How does the train get such a great amount of radians and still avoid collision? In reality, how much time will the train need to complete one revolution?
A:
If $v>1$ then the formula for $\theta$ is $$\theta=\frac{\pi}{2}-\arctan \frac{v-1}{\tan 45^\circ}$$
For $v=2$ the minimum speed is $v-1=1$, so that the minimum is $1$, but the train has to climb the hill.
If $v=1.25$ then the minimum speed is $1.25-1=0.25$, but then the train has to slow down at the top
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self.assertEqual([] or {“”, [] or “x”}, get_commands_in_ns(manager_ns, None))
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self.assertEqual(
[“fs.fsid”, “fs.path”, “fs.cache.dir”, “fs.size”, “fs.last.mod.time”],
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