PHY 140Y FOUNDATIONS OF PHYSICS Tutorial Questions #12 Solutions December 3/4

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1 PHY 40Y FOUNDTIONS OF PHYSICS Tutorial Questions # Solutions Deember 3/4 orentz Veloity ddition, Time Dilation, and orentz Contration. Two spaeships are traelling with eloities of 0.6 and 0.9 relatie to a third obserer. (a) What is the speed of one spaeship relatie to the other spaeship if they are going in the same diretion? (b) What is their relatie speed if they are going in opposite diretions? (a) Define referene frame for one of the spaeships, say the one traelling at 0.6 (all it #). Then the obserer moes at eloity 0.6 with respet to this referene frame u? - referene frame of spaeship ' - referene frame of the obserer spaeship - moes at u' 0.9 wrt ' The speed of spaeship # with respet to spaeship # is then gien by orentz eloity addition: u' u m/s u' + (0.9) (b) Now the spaeships are going in opposite diretions, so u u? - referene frame of spaeship ' - referene frame of the obserer spaeship - moes at u' -0.9 wrt ' PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page

2 orentz eloity addition gies: u' u m/s ( 0.6) + u' + ( 0.9) The negatie sign indiates that the spae ships are moing in opposite diretions.. roket moes towards a mirror at 0. relatie to the referene frame, shown below. The mirror is stationary relatie to. light pulse emitted by the roket traels to the mirror and is refleted bak to the roket. The front of the roket is. 0 m from the mirror (as measured in ) at the moment that the light pulse leaes the roket. What is the total trael time of the pulse as measured by an obserer in (a) referene frame, and (b) the front of the roket? Define D. 0 m dise from the front of the roket to the mirror 0. β 0. light pulse D mirror (a) In referene frame : trael time to the mirror: trael time bak to the roket: Thus the total trael time is: t D t D t D β t D t t + t + D β t D β t Rearrange and sole for t D ( + β) ( (. 0 m) + 0.)( m/s) t 6667 s.(5) hours (b) We an apply time dilation to alulate the trael time measured by an obserer on the front of the roket beause the two eents (light emitted from roket and light returned to roket) our at the same plae for this obserer (at the front of the roket). Define as the referene frame of the roket. PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page

3 Thus: t' t D D ( + β) β + β (. 0 m) m/s D + β + β D ( + β) 4000 s β.() hours 3. rod of proper length is at rest in referene frame F'. The rod lies in the (x', y') plane and makes an angle of with the x' axis. F' moes with a onst eloity parallel to the x-axis of another frame F. (a) What is the alue of if, as measured in F, the rod is at 45 to the x axis? (b) What is the length of the rod as measured in F under the situation gien in part (a)? Derie general expressions, and also look at the speial ase of sin - (3/5). y 45 o x F F' (a) In referene frame F : x omponent of length: x' os y omponent of length: y' sin y' x' We an apply length ontration to onert these to referene frame F: x omponent of length: x' os x (length ontration applies) y omponent of length: y y' sin (no motion in the y diretion) If the rod is at angle 45 to the x axis, as measured in F, then the x and y omponents must be equal. Note that this angle is not the same as in frame F. Thus: os sin os sin PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page 3

4 The general expression for the eloity in F is then: For the speial ase of sin - (3/5): sin - (3/5) 36.7 (in referene frame F ) and 3/4 So: 36.7 o m/s (b) The length of the rod, as measured in referene frame F, is: ' x' + y' sin sin os sin os ( sin ) From part (a), we hae, so the general expression is: ' sin + sin For the speial ase of sin - (3/5): ' sin 3 5 orentz Transformations 4. Spaeship of proper length is traelling east at speed, and spaeship B of proper length is traelling west at speed B, both as seen from Earth. The pilot of spaeship sets a lok to zero when the front of spaeship B passes by. (The spaeship pilots sit in the nose ones.) Use orentz transformations to derie an expression for the time at whih, aording to the pilot of spaeship, the tail of spaeship B passes by. Define referene frame S at rest with respet to spaeship, and referene frame S at rest with respet to spaeship B. Set up the x and x axes as shown below. Define the two eents and the releant x and x oordinates for these eents. PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page 4

5 Eent #: the front of spaeship B passes by the front of spaeship the front of is at x 0, the bak of is at x - in referene frame S the front of B is at x 0, the bak of B is at x in referene frame S set t t 0 as the time when eent ours and x x 0 EVENT B - 0' 0 B and B x' are measured w.r.t. Earth x Eent #: the bak of spaeship B passes by the front of spaeship the front of remains at x 0, and the bak of at x - in referene frame S the front of B remains at x, and the bak of B at x in referene frame S this ours at time t in referene frame S, when x 0 EVENT B 0' - B 0 and B x' are measured w.r.t. Earth x We need t apply the orentz transformations to find time when Eent ours. These require the relatie eloity of spaeship B with respet to, whih we first find using orentz eloity addition: u' + + B u B + u' + where u eloity of spaeship B with respet to spaeship eloity of spaeship with respet to Earth B eloity of spaeship B with respet to Earth PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page 5

6 Now, apply the orentz transformation for position: x' (x ut) (0 ut) ut t u But note that eloity u is in the negatie x diretion in referene frame S, so: t u This is the time for Eent as reorded by the pilot in spaeship. So pilot sees the length of spaeship B ontrated from to /. Beause spaeship B is moing at speed u relatie to spaeship, the time it takes for B to pass is the length of B (/) diided by the relatie speed u. Energy in STR 5. Solar energy reahes the Earth at the rate of about 400 W/m of surfae area perpendiular to the diretion to the Sun. By how muh does the mass of the Sun derease in eah seond? The mean radius of Earth's orbit is.5 0 km. Would the real mass loss of the Sun be greater than or less than your alulated answer, and why? The total energy output of the Sun (per unit time) is: E 400 W/m surfae area of a sphere whose radius is Earth' s orbit W/m W/m ( ) 4πR 4π(.50 0 W ( m/s) m) Joules/seond This amount of energy is released by the Sun eery seond, resulting in a loss of mass gien by: E m 6 E J 9 m kg So the Sun loses 4.4 billion kg per seond. The real mass loss is atually greater than this beause the Sun also emits the solar wind (whih onsists of harged partiles), whih add to the mass loss due to radiant energy just alulated. PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page 6

7 6. proton (mass kg) is moing at speed (a) What is the proton s total relatiisti energy? (b) What is the proton s kineti energy? () What is the proton s rest energy? (d) What is the magnitude of the proton s relatiisti momentum? (a) First ealuate the relatiisti fator: The proton s total relatiisti energy is: E m.9 ( Joules (b) The proton s kineti energy is: K m m ( )m (.9 ) ( Joules () The proton s rest energy is: E E K m o ( Joules kg) ( m / s) 7 kg) ( m / s) kg) ( m / s) (d) The magnitude of the proton s relatiisti momentum is: p m or.9 ( kg m / s E p E J m / s.03 0 kg m / s kg) ( m / s) PHY 40Y Foundations of Physis (K. Strong) Tutorial Solutions, page 7

Physics 43 HW 3 Serway Chapter 39 & Knight Chapter 37

Physics 43 HW 3 Serway Chapter 39 & Knight Chapter 37 Physis 43 HW 3 Serway Chapter 39 & Knight Chapter 37 Serway 7 th Edition Chapter 39 Problems: 15, 1, 5, 57, 60, 65 15. Review problem. An alien ivilization oupies a brown dwarf, nearly stationary relative