Physics 43 HW 3 Serway Chapter 39 & Knight Chapter 37

Transcription

1 Physis 43 HW 3 Serway Chapter 39 & Knight Chapter 37 Serway 7 th Edition Chapter 39 Problems: 15, 1, 5, 57, 60, Review problem. An alien ivilization oupies a brown dwarf, nearly stationary relative to the Sun, several lightyears away. The etraterrestrials have ome to love original broadasts of I Love Luy, on our television hannel, at arrier frequeny 57.0 MHz. Their line of sight to us is in the plane of the Earth s orbit. Find the differene between the highest and lowest frequenies they reeive due to the Earth s orbital motion around the Sun. P39.15 The orbital speed of the Earth is as desribed by F = ma: ( N m kg )( kg) Gm m S E E = r m v r Gm 4 v = S = =.9 10 m s. 11 r m The maimum frequeny reeived by the etraterrestrials is ( 4 ) ( ) ( ) ( ) 1+ v m s m s ( Hz) Hz. v.9 10 m s m s 6 6 obs = fsoure = = 4 f The minimum frequeny reeived is ( 4 ) ( ) ( ) ( ) v.9 10 m s m s ( Hz) Hz. 1+ v m s m s 6 6 obs = fsoure = = 4 f The differene, whih lets them figure out the speed of our planet, is 6 4 ( ) 10 Hz= Hz. 1. A physiist drives through a stop light. When he is pulled over, he tells the polie offier that the Doppler shift made the red light of wavelength 650 nm appear green to him, with a wavelength of 50 nm. The polie offier writes out a traffi itation for speeding. How fast was the physiist traveling, aording to his own testimony? P39.1 For the light as observed 1+ v 1+ v fobs = = fsoure = λ v vλ obs obs 7 soure 1+ v λsoure 650 nm 1+ v = = = 1.5 = 1.56 v λ 50 nm v v v v = = = v = 0.0 = m s

2 5. Ted and Mary are playing a game of ath in frame S, whih is moving at with respet to frame S, while Jim, at rest in frame S, wathes the ation. Ted throws the ball to Mary at 0.00 (aording to Ted) and their separation (measured in S ) is m. (a) Aording to Mary, how fast is the ball moving? (b) Aording to Mary, how long does it take the ball to reah her? () Aording to Jim, how far apart are Ted and Mary, and how fast is the ball moving? (d) Aording to Jim, how long does it take the ball to reah Mary? P39.5 (a) Sine Mary is in the same referene frame, S, as Ted, she measures the ball to have the same speed Ted observes, namely u = (b) L 1 p m 3 Δ t = = = s u m s ( ) v () L Lp ( ) ( 0.600) 1 1 = = m = m Sine v = and u = 0.00, the veloity Jim measures for the ball is u ( 0.00) ( 0.600) 1+ ( 0.00)( 0.600) u + v + = = = uv (d) Jim measures the ball and Mary to be initially separated by m. Mary s motion at and the ball s motion at 0.35 nibble into this distane from both ends. The gap loses at the rate = 0.95, so the ball and ather meet after a time m 3 Δ t = = s m s ( ) Imagine that the entire Sun ollapses to a sphere of radius Rg suh that the work required to remove a small mass m from the surfae would be equal to its rest energy m. This radius is alled the gravitational radius for the Sun. Find Rg. (It is believed that the ultimate fate of very massive stars is to ollapse beyond their gravitational radii into blak holes.) P39.60 If the energy required to remove a mass m from the surfae is equal to its rest energy m, GM then s m = m R and g GM R g = = ( N m kg )( kg) ( m s) s 3 R = m = 1.47 km g

3 57. An alien spaeship traveling toward the Earth launhes a landing raft with an advane guard of purhasing agents and physis teahers. The lander travels in the same diretion with a speed of 0.00 relative to the mother ship. As observed on the Earth, the spaeship is 0.00 ly from the Earth when the lander is launhed. (a) What speed do the Earth observers measure for the approahing lander? (b) What is the distane to the Earth at the time of lander launh, as observed by the aliens? () How long does it take the lander to reah the Earth as observed by the aliens on the mother ship? (d) If the lander has a mass of kg, what is its kineti energy as observed in the Earth referene frame? P39.57 (a) Take the spaeship as the primed frame, moving toward the right at v = Then u =+ 0.00, and u u + v = = = 1+ ( uv ) 1+ ( 0.00)( 0.600) (b) () L L p = : ( ) ( ) γ L = 0.00 ly = ly The aliens observe the ly distane losing beause the probe nibbles into it from one end at 0.00 and the Earth redues it at the other end at Thus, ly time = = yr (d) 1 K = 1 m : u 1 K = kg m s 1 ( ) K = J 5 ( )( )

4 65. Suppose our Sun is about to eplode. In an effort to esape, we depart in a spaeraft at v = 0.00 and head toward the star Tau Ceti, 1.0 ly away. When we reah the midpoint of our journey from the Earth, we see our Sun eplode and, unfortunately, at the same instant we see Tau Ceti eplode as well. (a) In the spaeraft s frame of referene, should we onlude that the two eplosions ourred simultaneously? If not, whih ourred first? (b) What If? In a frame of referene in whih the Sun and Tau Ceti are at rest, did they eplode simultaneously? If not, whih eploded first? P39.65 We hoose to write down the answer to part (b) first. (b) Consider a hermit who lives on an asteroid halfway between the Sun and Tau Ceti, stationary with respet to both. Just as our spaeship is passing him, he also sees the blast waves from both eplosions. Judging both stars to be stationary, this observer onludes that the tw o stars blew up simultaneously. (a) We in the spaeship moving past the hermit do not alulate the eplosions to be simultaneous. We measure the distane we have traveled from the Sun as v L = Lp = ( 6.00 ly ) ( 0.00) = 3.60 ly We see the Sun flying away from us at 0.00 while the light from the Sun approahes at Thus, the gap between the Sun and its blast wave has opened at 1.0, and the time we alulate to have elapsed sine the Sun eploded is 3.60 ly.00 yr 1.0 = We see Tau Ceti as moving toward us at 0.00, while its light approahes at 1.00, only 0.00 faster. We measure the gap between that star and its blast wave as 3.60 ly and growing at We alulate that it must have been opening for 3.60 ly 1.0 yr 0.00 = and onlude that Tau Ceti eploded 16.0 years before the Sun.

5 Knight nd Edition Chapter 37 Eerises & Problems: 1, 13, 15, 54, 61, You are standing at = 9.0km.Lightning bolt 1 strikes at = 0 km and lightning bolt strikes at 1.0km. Both flashes reah your eye at the same time. Your assistant is standing at = 3.0km. Does your assistant see the flashes at the same time? If not, whih does she see first and what is the time differene between the two? Model: You and your assistant are in the same referene frame. Light from the two lightning bolts travels toward you and your assistant at 300 m/μs. You and your assistant have synhronized loks. flashes reah your eye means that you see the lightning strikes as simultaneous but sine they are at different distanes, you know that they struk at different times. Visualize: Solve: Bolt 1 is 9.0 km away, so it takes 30 μs for the light to reah you ( 9000 m 300 m/ μs). Bolt is 3.0 km away from you, so it takes 10 μs to reah you. Sine both flashes reah your eye at the same time, event 1 happened 0 μs before event. If event 1 happened at time t 1 = 0 then event happened at time t = 0 μs. For your assistant, it takes light from bolt 1 10 μs to reah her and light from bolt 30 μs to reah her. She sees the flash from bolt 1 at t = 10 μs and the flash from bolt at t = 50 μs. That is, your assistant sees flash 40 μs after she sees flash You are standing at = 9.0km.Lightning bolt 1 strikes at = 0 km and lightning bolt strikes at 1.0km. You see the flash from bolt at t= 10 μs and the flash from bolt 1 at t = 50μs. Your assistant is standing at = 3.0km. Does your assistant see the flashes as simultaneous? If not, whih does she see first and what is the time differene between the two? Model: You and your assistant are in the same referene frame. Light from the two lightning bolts travels toward you and your assistant at 300 m/μs. You and your assistant have synhronized loks. Visualize: same as above Solve: Bolt 1 hits 9.0 km away, so the light takes 30 µs to reah you (9000 m 300 m/μs). You see this flash at t = 50 μs, so the lightning hit at t 1 = 0 μs. Light from bolt, whih hits 3.0 km away, takes 10 μs to reah you. You see it at 10 μs, so the lightning hit at t = 0 μs. The strikes are not simultaneous. Bolt hits first, 0 μs before bolt 1. Your assistant is in your inertial referene frame, so your assistant agrees that bolt hits first, 0 μs before bolt 1. Assess: A simple alulation would show that your assistant sees the flashes at the same time. When the flashes are seen is not the same as when the events happened Model: Your personal roket raft is an inertial frame moving at 0.9 relative to stars A and B. Solve: In your frame, star A is moving away from you and star B is moving toward you. When you are eatly halfway between them, both the stars eplode simultaneously. The flashes from the two stars travel toward you with speed. Beause (i) you are at rest in your frame, (ii) the eplosions are equally distant, and (iii) the light speed is, independent of the fat that the stars are moving in your frame, the light will arrive simultaneously.

6 Model: Let the earth be frame S and the roket be frame S. S moves with speed v relative to S. Solve: (a) The round-trip distane is 60 ly. If the roket takes time t to make the round trip, as measured on earth, its speed (as a fration of ) is v 60 ly 60 yr = = Δt Δt where we used = 1 ly/yr (1 light year per year). The astronaut s elapsed time t is the proper time, so τ = 0 yr. The time dilation equation is Δτ 0 yr Δ t = = 1 (60 yr/ Δ t) = (0 yr/ Δt) 1 ( v / ) 1 (60 yr/ Δt) Solving for t gives t = y, and thus v 60 y v = y = = (b) The roket starts with rest energy E i = m and aelerates to have energy E f = γ p m. Thus the energy needed to aelerate the roket is E = E f E 1 = (γ p 1)m This is just the kineti energy K gained by the roket. We know the roket s speed, so 1 Δ E = 1 (0,000 kg)( m/s) = J 1 ( ) () The total energy used by the United States in 000 was J. To aelerate the roket would require roughly 760 times the total energy used by the United States Model: Let S be the ground s referene frame and S the muon s referene frame. S travels with a speed of v relative to S. Solve: (a) The half-life of a muon at rest is 1.5 μs. That is, the half-life in the muon s rest frame S is 1.5 μs. So, Δt = Δτ = 1.5 μs. The half-life of 7.5 μs, when muons have been aelerated to very high speed, means that Δt = 7.5 μs. Thus (b) The muon s total energy is Δτ 1.5 μs Δ t = 7.5 μs = = v = 0.0 v = 0.9 v ( v) ( )( )( ) 11 E = γ pm = m = kg m/s.5 10 J = v Model: Mass and energy are equivalent and given by Equation Solve: (a) The sun radiates energy for s per year. The amount of energy radiated per year is Sine E 0 = m, the amount of mass lost is ( J/s)( s) = J/y 34 E J m = = = 0 ( m/s) kg kg (b) Sine the mass of the sun is kg, the sun loses % of its mass every year. () The lifetime of the sun an be estimated to be kg kg/y T = = years The sun will not really last this long in its urrent state beause fusion only takes plae in the ore and it will beome a red giant when the ore hydrogen is all fused.