37 1 Model S and S are inertial frames that overlap at t 0 Frame S moves with a speed v 5 0 m s along the x direction relative to frame S Visualize The figure shows a pictorial representation of the S and S frames at t 1 0 s and 5 0 s Solve From the figure the observer in S finds the position of the first explosion at x 1 5 0 m at t 1 0 s The 5 0 m at t 5 0 s We can get the same answers using the Galilean position of the second explosion is transformations of position x 2 vt 10 m 5 0 m s 1 0 s 5 0 m at 1 0 s x 1 x 1 x 2 x 2 vt 20 m 5 0 m s 5 0 s 5 0 m at 5 0 s 37 2 Model S and S are inertial frames S moves relative to S with speed v Solve a Using the Galilean transformations of position x 1 x1 vt1 4 0 m x1 v 1 0 s x1 4 0 m v 1 0 s x 2 x 2 vt 2 4 0 m x2 v 3 0 s x2 4 0 m v 3 0 s Because x1 x2 b The positions of the two explosions in the S frame are 4 0 m v 1 0 s 4 0 m v 3 0 s v 4 0 m s x1 4 0 m 4 0 m s 1 0 s 8 0 m x2 4 0 m 4 0 m s 3 0 s 8 0 m 37 3 Model S is the ground s frame of reference and S is the sprinter s frame of reference Frame S moves relative to frame S with speed v Visualize Solve The speed of a sound wave is measured relative to its medium The medium is still air on the ground which is our frame S The sprinter travels to the right with reference frame S at velocity v Using the Galilean transformations of velocity u 1 360 m s u 1 v v sound v u 2 330 m s u v v sound 2 v Adding the two above equations From the first equation 30 m s 2v vsprinter 15 m s 360 m s vsound 15 m s vsound 345 m s Assess Notice that the Galilean transformations use velocities and not speeds It is for that reason 1 360 m s u 37 4 Model You are on the ground in frame S and the baseball pitcher is in the pickup in frame S S moves relative to S with velocity v Visualize The figure shows a pictorial representation of the two frames The Galilean transformation uses velocities not speeds so u and u are negative Solve The speed of the baseball in the two frames is u 40 m s and u 10 m s From Equation 37 2 u u v v u u 10 m s 40 m s 30 m s 37 5 Model The boy on a bicycle is frame S and the ground is frame S S moves relative to S with a speed v 5 0 m s The frames S and S overlap at t 0 Visualize The figure shows a pictorial representation of the two frames a When the newspaper is thrown forward Solve u x 8 0 m s The Galilean transformation of velocity is b When the newspaper is thrown backward u x 8 0 m s In this case ux u x v 8 0 m s 5 0 m s 13 m s ux u x v 8 0 m s 5 0 m s 3 0 m s Thus the speed is 3 0 m s c When the newspaper is thrown to the side u y u y 8 0 m s Also u x u x v 0 m s 5 0 m s 5 0 m s Thus the newspaper s speed is 5 0 m s u u u 2 2 x 2 y 8 0 m s 9 4 m s 2 37 6 Model Assume the spacecraft is an inertial reference frame Solve Light travels at speed c in all inertial reference frames regardless of how the reference frames are moving with respect to the light source Relative to the spacecraft the starlight is approaching at the speed of light c 3 00 108 m s 37 7 Model Assume the starship and the earth are inertial reference frames It has been found that light travels at 3 00 108 m s in every inertial frame regardless of how the reference Solve frames are moving with respect to each other An observer on the earth will measure the laser beam s speed as 3 00 108 m s 37 8 Model Assume the earth is an inertial reference frame Solve Light travels at speed c in all inertial reference frames regardless of their motion with respect to the light source The speed of each photon will be c in any such inertial reference frame 37 9 Model The clocks are in the same reference frame Visualize Solve The speed of light is c 300 m s 0 30 m ns The distance from the origin to the point x y z 30 m 40 m 0 m is 30 m 2 40 m 2 50 m So the time taken by the light to travel 50 m is The clock should be preset to 167 ns 50 m 0 30 m ns 167 ns 37 10 Model Bjorn and firecrackers 1 and 2 are in the same reference frame Light from both firecrackers travels towards Bjorn at 300 m s Visualize Solve Bjorn is 600 m from the origin Light with a speed of 300 m s takes 2 0 s to reach Bjorn Since this flash reaches Bjorn at t 3 0 s it left firecracker 1 at t1 1 0 s The flash from firecracker 2 takes 1 0 s to reach Bjorn So the light left firecracker 2 at t2 2 0 s Note that the two events are not simultaneous although Bjorn sees the events as occurring at the same time 37 11 Model Bianca and firecrackers 1 and 2 are in the same reference frame Light from both firecrackers travels toward Bianca at 300 m s Visualize Solve The flash from firecracker 1 takes 2 0 s to reach Bianca exploded at t1 1 0 s because it reached Bianca s eye at 3 0 s The flash from the firecracker 2 takes 1 0 s to reach Bianca Since firecrackers 1 and 2 exploded simultaneously the explosion occurs at t2 1 0 s So the light from firecracker 2 reaches Bianca s eye at 2 0 s Although the events are simultaneous Bianca sees them occurring at different times The firecracker 600 m 300 m s 37 12 Model You and your assistant are in the same reference frame Light from the two lightning bolts travels toward you …
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