2015 USA Physics Olympiad Exam Part A 4 ii.Numerically determine the minimum energy of a bouncing neutron.The mass of a neutron is mn=1.675x 10-27 kg=940 MeV/c2;you may express your answer in either Joules or eV. iii.Determine the bounce height of one of these minimum energy neutrons. Solution The integral is not particular difficult to solve, H 2v2m VE-mgx dx; 2V2mE V1-mgx/E da, Jo 2V2mE E V1-u du, mg Jo E 2V2mE vu dv, mg Jo =2v2 E3/22 mg3 So 9mg2h2 1 2/3 En= 32 n+2) Solving for the minimum energy we get E0= 9mg2h2 9(mc2)g2h2 =1.1×10-12eV 128 128c2 The bounce height is given by H= E0=10μm mg This is a very measurable distance! c.Let Eo be the minimum energy of the bouncing neutron and f be the frequency of the bounce. Determine an order of magnitude estimate for the ratio E/f.It only needs to be accurate to within an order of magnitude or so,but you do need to show work! Solution Asf=l/△t and Eo≈△E,Heisenberg's uncertainty relation yeilds △E△t≈h Copyright C2015 American Association of Physics Teachers2015 USA Physics Olympiad Exam Part A 4 ii. Numerically determine the minimum energy of a bouncing neutron. The mass of a neutron is mn = 1.675×10−27 kg = 940 MeV/c 2 ; you may express your answer in either Joules or eV. iii. Determine the bounce height of one of these minimum energy neutrons. Solution The integral is not particular difficult to solve,  n + 1 2  h = 2 Z H 0 p dx = 2√ 2m Z H 0 p E − mgx dx, = 2√ 2mE Z H 0 p 1 − mgx/E dx, = 2√ 2mE E mg Z 1 0 √ 1 − u du, = 2√ 2mE E mg Z 1 0 √ v dv, = 2√ 2 E3/2 √ mg 2 3 so En = 3 r 9mg2h 2 32  n + 1 2 2/3 Solving for the minimum energy we get E0 = 3 r 9mg2h 2 128 = 3 r 9(mc2)g 2h 2 128c 2 = 1.1 × 10−12eV The bounce height is given by H = E0 mg = 10 µm This is a very measurable distance! c. Let E0 be the minimum energy of the bouncing neutron and f be the frequency of the bounce. Determine an order of magnitude estimate for the ratio E/f. It only needs to be accurate to within an order of magnitude or so, but you do need to show work! Solution As f = 1/∆t and E0 ≈ ∆E, Heisenberg’s uncertainty relation yeilds ∆E∆t ≈ h Copyright c 2015 American Association of Physics Teachers