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2018-04-30 Intermediate

         

BB is the area of the blue region, and RR 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 4 m/s2.\SI{4}{m/s^2}. As soon as the first car reaches a speed of 6 m/s,\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 6 m/s,\SI{6}{m/s}, and accelerates at 4 m/s2\SI{4}{m/s^2} until it reaches the speed 20 m/s.\SI{20}{m/s}.

When all cars have reached the same speed 20 m/s,\SI{20}{m/s}, how far apart are two adjacent cars (in meters)?

For two random positive real numbers x x and y y chosen uniformly and independently from the interval (1,1000)(1,1000), determine the probability that x+y=x+y. \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 mrv˙r=uem˙r,m_\textrm{r}\,\dot{v}_\textrm{r}= u_\textrm{e}\, \dot{m}_\textrm{r}, where ueu_\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, ueu_\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 ueu_\textrm{e} for the combustion of methane (CHX4),(\ce{CH4}), the propellant used by SpaceX for its Raptor rocket engine program.

Details and Assumptions:

  • The specific enthalpy of combustion for methane is ΔcH890 kJ/mol.\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 COX2+4HX2CHX4+2HX2O.\ce{CO2 + 4H2 -> CH4 + 2H2O}.

(x1)(x22)(x33)(x2020) \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 x203x^{203} in the expansion of this expression?

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