2015 USA Physics Olympiad Exam Part A 2 Part A Question Al Consider a particle of mass m that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity.The total mechanical energy of the particle is E and the acceleration of free fall is g.Treat the particle as a point mass and assume the motion is non-relativistic. a.An estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce.Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle,determine the value of the energy Ea where quantum effects become important.Write your answer in terms of some or all of g,m,and Planck's constant h. Solution Full points will only be awarded if it is clear that the examinee knew the deBroglie wavelength expression. The deBroglie wavelength is p=h/ so if the height H of the bounce is given by E mgH 2m and入=H,then h2 mgH= 2mH2 or H3= h2 2m2g or 瓦=5mg22 An answer of mg2h2 is acceptable,but will not receive full points if it was derived by dimensional analysis alone. b.A second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle.Assuming that the particle rises to a height H,we can write -(+ H where p is the momentum as a function of height r above the ground,n is a non-negative integer,and h is Planck's constant. i.Determine the allowed energies En as a function of the integer n,and some or all of g, m,and Planck's constant h. Copyright C2015 American Association of Physics Teachers2015 USA Physics Olympiad Exam Part A 3 Part A Question A1 Consider a particle of mass m that elastically bounces off of an infinitely hard horizontal surface under the influence of gravity. The total mechanical energy of the particle is E and the acceleration of free fall is g. Treat the particle as a point mass and assume the motion is non-relativistic. a. An estimate for the regime where quantum effects become important can be found by simply considering when the deBroglie wavelength of the particle is on the same order as the height of a bounce. Assuming that the deBroglie wavelength is defined by the maximum momentum of the bouncing particle, determine the value of the energy Eq where quantum effects become important. Write your answer in terms of some or all of g, m, and Planck’s constant h. Solution Full points will only be awarded if it is clear that the examinee knew the deBroglie wavelength expression. The deBroglie wavelength is p = h/λ so if the height H of the bounce is given by E = mgH = p 2 2m and λ = H, then mgH = h 2 2mH2 or H3 = h 2 2m2g or Eq = 3 r 1 2 mg2h 2 An answer of p3 mg2h 2 is acceptable, but will not receive full points if it was derived by dimensional analysis alone. b. A second approach allows us to develop an estimate for the actual allowed energy levels of a bouncing particle. Assuming that the particle rises to a height H, we can write 2 Z H 0 p dx =  n + 1 2  h where p is the momentum as a function of height x above the ground, n is a non-negative integer, and h is Planck’s constant. i. Determine the allowed energies En as a function of the integer n, and some or all of g, m, and Planck’s constant h. Copyright c 2015 American Association of Physics Teachers