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2018-04-30 Intermediate

         

\(B\) is the area of the blue region, and \(R\) is the area of the red region.

Which statement is correct?

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)?

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.

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:

  • The specific enthalpy of combustion for methane is \(\Delta_c H \approx \SI[per-mode=symbol]{890}{\kilo\joule\per\mole}.\)
  • Assume that all the energy released in combustion goes into the kinetic energy of the gas.
  • Beyond its performance, methane is nice because it could potentially be produced on other planets via the Sabatier reaction \(\ce{CO2 + 4H2 -> CH4 + 2H2O}.\)

\[ \big(x - 1\big)\big(x^2 - 2\big)\big(x^3 - 3\big)\cdots \big(x^{20} - 20\big)\]

What is the coefficient of \(x^{203}\) in the expansion of this expression?

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