\(B\) 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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