2014 USA Physics Olympiad Exam Part A 6 c.A typical room has a value of k 173 W/C.If the outside temperature is 40C,what minimum power should the air conditioner have to get the inside temperature down to 25C? Solution Don't forget to convert to Kelvin! From above, P= k(△T)2 T So P=130W Copyright C2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 6 c. A typical room has a value of k = 173 W/ ◦C. If the outside temperature is 40◦C, what minimum power should the air conditioner have to get the inside temperature down to 25◦C? Solution Don’t forget to convert to Kelvin! From above, P = k(∆T) 2 TL , so P = 130 W Copyright c 2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 7 Question A3 When studying problems in special relativity it is often the invariant distance As between two events that is most important,where As is defined by (△s)2=(c△t)2-[(△x)2+(△g)2+(△z)2] where c=3 x 108 m/s is the speed of light.1 a.Consider the motion of a projectile launched with initial speed vo at angle of 0o above the horizontal.Assume that g,the acceleration of free fall,is constant for the motion of the projectile. i.Derive an expression for the invariant distance of the projectile as a function of time t as measured from the launch,assuming that it is launched at t=0.Express your answer as a function of any or all of 0o,vo,c,g,and t. ii.The radius of curvature of an object's trajectory can be estimated by assuming that the trajectory is part of a circle,determining the distance between the end points,and measuring the maximum height above the straight line that connects the endpoints.As- suming that we mean "invariant distance"as defined above,find the radius of curvature of the projectile's trajectory as a function of any or all of 00,vo,c,and g.Assume that the projectile lands at the same level from which it was launched,and assume that the motion is not relativistic,so vo<c,and you can neglect terms with v/c compared to terms without. Solution The particle begins at ct=x=y=0 and takes a path satisfying x vot cos0o 1 z=vot sin 0o- 9 Thus (-(wotcos0)-(otsin which can be simplified to 2=(c2-喝2+5 sino+ 9 b.The particle reaches the ground again at t与-20sin0 If vo cos Otf 对=0 We are using the convention used by Einstein Copyright C2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 7 Question A3 When studying problems in special relativity it is often the invariant distance ∆s between two events that is most important, where ∆s is defined by (∆s) 2 = (c∆t) 2 − (∆x) 2 + (∆y) 2 + (∆z) 2 where c = 3 × 108 m/s is the speed of light.1 a. Consider the motion of a projectile launched with initial speed v0 at angle of θ0 above the horizontal. Assume that g, the acceleration of free fall, is constant for the motion of the projectile. i. Derive an expression for the invariant distance of the projectile as a function of time t as measured from the launch, assuming that it is launched at t = 0. Express your answer as a function of any or all of θ0, v0, c, g, and t. ii. The radius of curvature of an object’s trajectory can be estimated by assuming that the trajectory is part of a circle, determining the distance between the end points, and measuring the maximum height above the straight line that connects the endpoints. Assuming that we mean “invariant distance” as defined above, find the radius of curvature of the projectile’s trajectory as a function of any or all of θ0, v0, c, and g. Assume that the projectile lands at the same level from which it was launched, and assume that the motion is not relativistic, so v0 c, and you can neglect terms with v/c compared to terms without. Solution The particle begins at ct = x = y = 0 and takes a path satisfying x = v0t cos θ0 z = v0tsin θ0 − 1 2 gt2 Thus s 2 = (ct) 2 − (v0t cos θ0) 2 − (v0tsin θ0 − 1 2 gt2 ) 2 which can be simplified to s 2 = (c 2 − v 2 0 )t 2 + 1 2 gv0 sin θ0 t 3 + 1 4 g 2 t 4 b. The particle reaches the ground again at tf = 2 v0 sin θ g xf = v0 cos θtf zf = 0 1We are using the convention used by Einstein Copyright c 2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 8 and so the invariant distance between the endpoints is s2=(ct)2+(vocos0t)2 s≈2c20sin0 The maximum height above the ground is (vosin0)2 2g Because zmar <s,we have from similar triangles zma≈ 1s2 R≈ 4 Zmax R≈2 c.A rocket ship far from any gravitational mass is accelerating in the positive x direction at a constant rate g,as measured by someone inside the ship.Spaceman Fred at the right end of the rocket aims a laser pointer toward an alien at the left end of the rocket.The two are separated by a distance d such that dg<c2;you can safely ignore terms of the form(dg/c2)2 i.Sketch a graph of the motion of both Fred and the alien on the space-time diagram provided in the answer sheet.The graph is not meant to be drawn to scale.Note that t and x are reversed from a traditional graph.Assume that the rocket has velocity v=0 at time t=0 and is located at position x=0.Clearly indicate any asymptotes,and the slopes of these asymptotes. Solution Since the rocket-ship can never exceed the speed of light,yet it is always accelerating (in the local frame),it must approach an asymptote that has a slope of one on the space-time diagram shown.There is a slight challenge to consider,however.Since the rocket ship is an extended object,do the two ends(represented by Fred and the Alien) approach the same asymptote,or two different asymptotes? At this point we must remember a consequence of special relativity for a ship moving at relativistic speeds:the ship will contract in length as measured in the original frame. As the speed of the ship approaches that of light,the length of the ship will approach zero.The only way for that to happen is for the two ends of the ship to have slightly different accelerations. Copyright C2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 8 and so the invariant distance between the endpoints is s 2 = (ctf ) 2 + (v0 cos θ tf ) 2 s ≈ 2c v0 sin θ g The maximum height above the ground is zmax = (v0 sin θ) 2 2g Because zmax s, we have from similar triangles zmax 1 2 s ≈ 1 2 s R R ≈ 1 4 s 2 zmax R ≈ 2 c 2 g c. A rocket ship far from any gravitational mass is accelerating in the positive x direction at a constant rate g, as measured by someone inside the ship. Spaceman Fred at the right end of the rocket aims a laser pointer toward an alien at the left end of the rocket. The two are separated by a distance d such that dg c 2 ; you can safely ignore terms of the form (dg/c2 ) 2 . i. Sketch a graph of the motion of both Fred and the alien on the space-time diagram provided in the answer sheet. The graph is not meant to be drawn to scale. Note that t and x are reversed from a traditional graph. Assume that the rocket has velocity v = 0 at time t = 0 and is located at position x = 0. Clearly indicate any asymptotes, and the slopes of these asymptotes. Solution Since the rocket-ship can never exceed the speed of light, yet it is always accelerating (in the local frame), it must approach an asymptote that has a slope of one on the space-time diagram shown. There is a slight challenge to consider, however. Since the rocket ship is an extended object, do the two ends (represented by Fred and the Alien) approach the same asymptote, or two different asymptotes? At this point we must remember a consequence of special relativity for a ship moving at relativistic speeds: the ship will contract in length as measured in the original frame. As the speed of the ship approaches that of light, the length of the ship will approach zero. The only way for that to happen is for the two ends of the ship to have slightly different accelerations. Copyright c 2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 9 slope Assumies racket has "averase acceleratonofg but acceleratonis oi fred and alren will der sghtly. Rockct stays whole Flash 2 t=0 rFlash Alien If you had assumed(somewhat incorrectly)that the two ends of the ship have the same acceleration,then the two trajectories would be approaching two different asymptotes separated by a constant horizontal distance.But this would mean the apparent length of the ship was constant,regardless of speed.In the instantaneous rest frame of the ship we then require that Fred and the Alien be moving apart.This means that the ship must be stretching,and eventually breaking. Copyright C2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 9 If you had assumed (somewhat incorrectly) that the two ends of the ship have the same acceleration, then the two trajectories would be approaching two different asymptotes separated by a constant horizontal distance. But this would mean the apparent length of the ship was constant, regardless of speed. In the instantaneous rest frame of the ship we then require that Fred and the Alien be moving apart. This means that the ship must be stretching, and eventually breaking. Copyright c 2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 10 assumes Fred Alen experience same acceleration,but the rocket will eventually cip apart 4444444444n4 444444444444 Flash2 t=0 Flash I Alien 星 ii.If the frequency of the laser pointer as measured by Fred is f1,determine the frequency of the laser pointer as observed by the alien.It is reasonable to assume that fi>c/d. Solution If the spaceship is uniformly accelerating then we can choose a reference frame which is instantaneously at rest with respect to the spaceship at t=0. Consider two instantaneous flashes from the astronaut.Flash 1 is emitted by Fred at t=0,flash 2 is emitted by the Fred at t=T. Copyright C2014 American Association of Physics Teachers
2014 USA Physics Olympiad Exam Part A 10 ii. If the frequency of the laser pointer as measured by Fred is f1, determine the frequency of the laser pointer as observed by the alien. It is reasonable to assume that f1 c/d. Solution If the spaceship is uniformly accelerating then we can choose a reference frame which is instantaneously at rest with respect to the spaceship at t = 0. Consider two instantaneous flashes from the astronaut. Flash 1 is emitted by Fred at t = 0, flash 2 is emitted by the Fred at t = τ1. Copyright c 2014 American Association of Physics Teachers