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$B$ is the area of the blue region, and $R$ is the area of the red region.
Which statement is correct?
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The distance between any two adjacent cars at a red light is 2 meters.
As soon as the light turns green, the first car starts accelerating at $\SI{4}{m/s^2}.$ As soon as the first car reaches a speed of $\SI{6}{m/s},$ the second car starts accelerating at the same rate. In exactly the same fashion, each successive car starts accelerating when the car in front of it reaches $\SI{6}{m/s},$ and accelerates at $\SI{4}{m/s^2}$ until it reaches the speed $\SI{20}{m/s}.$
When all cars have reached the same speed $\SI{20}{m/s},$ how far apart are two adjacent cars (in meters)?
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For two random positive real numbers $x$ and $y$ chosen uniformly and independently from the interval $(1,1000)$, determine the probability that $\lfloor x+y \rfloor = \lfloor x \rfloor + \lfloor y \rfloor.$
Notation: $\lfloor \cdot \rfloor$ denotes the floor function.
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The equation of motion for rockets is $m_\textrm{r}\,\dot{v}_\textrm{r}= u_\textrm{e}\, \dot{m}_\textrm{r},$ where $u_\textrm{e}$ is the speed of the exhaust shooting out the back. The final velocity of the rocket is a few multiples of the exhaust speed, so it's quite important, but in physics textbooks, $u_\textrm{e}$ is usually treated like a freely adjustable parameter whose origin and precise value is a detail left for engineers to worry about. With a little bit of approximation, we can do better than that.
Estimate $u_\textrm{e}$ for the combustion of methane $(\ce{CH4}),$ the propellant used by SpaceX for its Raptor rocket engine program.
Details and Assumptions:
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$\big(x - 1\big)\big(x^2 - 2\big)\big(x^3 - 3\big)\cdots \big(x^{20} - 20\big)$
What is the coefficient of $x^{203}$ in the expansion of this expression?
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