SPH4U UNIVERSITY PHYSICS REVOLUTIONS IN MODERN PHYSICS:... L (P.588-591) Special Relatiity Time dilation is only one of the consequences of Einstein s special theory of relatiity. Since reference frames also inole measurements of position and length, there are also consequences that deal with space, such as the contraction, or compression, of length. December 16, 01 4U5-1 Special Relatiity As well as changing our understanding of time and length, special relatiity also changes our understanding of momentum, energy, and mass. Despite the short-comings of some of the older ways of thinking about these concepts, they are still useful in many situations. Science is a process; sometimes there are incremental changes, and sometimes new information forces a complete rethinking of what we know. December 16, 01 4U5-1
Length Measurements Recall the example of obserer 1 on the railway car and obserer on the ground near the tracks. Suppose obserer marks two locations, A and B, on the ground. She then measures these locations to be a distance L s apart on the x-axis. Consider how obserers 1 and might each measure this length or distance. December 16, 01 4U5-3 Length & Moing Reference Frame Obserer 1 measures the distance between points A and B by using a clock to measure the time, )t s, it takes him to trael between the two points, together with his known speed,. This is the proper time interal ()t s ) because obserer 1, who measures the start and finish times, is stationary relatie to the clock. December 16, 01 4U5-4 Length & Moing Reference Frame In this case, the distance measured by obserer 1 is L m = )t s where L m is the relatiistic length of an object or the distance between two points as measured by an obserer who is moing relatie to the object or distance. RELATIVISTICLENGTH (L m ) the length as measured by an obserer who is moing with speed relatie to the object or distance December 16, 01 4U5-5
Length & Stationary Reference Frame When obserer measures with her clock how long it takes for obserer 1 to trael from A to B, the alue she determines for )t m is: t = m t s 1- c December 16, 01 4U5-6 Length & Stationary Reference Frame In this case, the distance measured by obserer is L s = )t m where L s is the proper length of an object or the distance between two points as measured by an obserer who is stationary relatie to the object or distance. PROPERLENGTH (L s ) the length as measured by an obserer who is stationary relatie to the object or distance December 16, 01 4U5-7 Multiplying both sides of the time dilation equation by and then substituting the two length measurements (L s = )t m and L m = )t s ) into the time dilation equation gies the following results: L t = ms L t mss 1- c which can then be rearranged to gie: L =L m s 1- c December 16, 01 4U5-8 3
Because )t m is different from )t s due to time dilation, it only makes sense that the lengths measured by the two obserers will also be different. In this case, the length L m measured by obserer 1 is shorter than the length L s measured by obserer. This effect, called length contraction, or compression, is the shortening of distances in an inertial frame of reference moing relatie to an obserer in another inertial frame of reference. Length contraction is the spatial counterpart to time dilation. December 16, 01 4U5-9 Length contraction only occurs along the direction of motion. For example, a cylindrical spaceship moing past the Earth at a ery high speed would appear shorter from tip to tail (but of the same diameter) due to length contraction. December 16, 01 4U5-10 LENGTH CONTRACTION the shortening of distances in one inertial FOR that is moing relatie to an obserer in another inertial FOR (i.e. L s $ L m ) only occurs along the direction of motion is the spatial counterpart to time dilation (i.e. eents that look smaller last longer) L = m Ls 1- c where L m L s is the relatiistic length of the object/distance (m) is the proper length of the object/distance (m) December 16, 01 4U5-11 4
1. If the equation for length contraction is true, why hae we not noticed length contraction before now? For the same reason time dilation is not noticeable at ordinary speeds, is much smaller than c, and is een smaller than c. Therefore, for speeds that are small compared to c, the fraction L m /L s is nearly 1. December 16, 01 4U5-1. A spacecraft passes Earth at a speed of.00 x 10 8 m/s. If obserers on Earth measure the length of the spacecraft to be 554 m, how long would it be according to its passengers? L s = 743 m December 16, 01 4U5-13 3. An asteroid has a length of 75 km. A rocket passes by parallel to the long axis at a speed of 0.50c. What will be the length of the asteroid as measured by obserers in the rocket? L m = 70 km December 16, 01 4U5-14 5
4. A spacecraft passes a spherical space station. Obserers in the spacecraft see the station s minimum diameter as 65 m and the maximum diameter as 35 m. (a) How fast is the spacecraft traelling relatie to the space station? (a) = 1.74 x 10 8 m/s (1.736...) December 16, 01 4U5-15 4. A spacecraft passes a spherical space station. Obserers in the spacecraft see the station s minimum diameter as 65 m and the maximum diameter as 35 m. (b) Why does the station not look like a sphere to the obserers in the spacecraft? (b) length contraction only occurs along the direction of motion December 16, 01 4U5-16 Muons & Eidence for & The decay of unstable elementary particles called muons demonstrates how length contraction and time dilation complement each other. One source of muons is the cosmic radiation that collides with atoms in Earth s upper atmosphere. According to Newtonian mechanics, most of these muons should decay after traelling about 660 m into the atmosphere. Yet experimental eidence shows that a large number of muons decay after traelling 4800 m oer seen times as far. Why does this happen? Muons are particles that are about 07 times as massie as electrons, trael at speeds of about 0.99c, and decay in. ms for an obserer at rest relatie to the muons. December 16, 01 4U5-17 6
Muons & Eidence for & The only know explanation comes from special relatiity. Consider Earth as the stationary frame of reference. As obsered from Earth, these muons undergo time dilation. They also undergo length contraction, but they are so small to begin with that this is a minor effect. Due to time dilation at ery high speeds, the muons clocks run slower relatie to Earth clocks, so their lifetimes as measured on Earth increase by a factor of seen. This allows them to trael the greater distance. December 16, 01 4U5-18 Muons & Eidence for & 5. What would this situation look like in the muon frame of reference? they would still decay after. ms but the 4800 m they trael (in our frame of reference) shortens to 660 m (in their frame of reference) December 16, 01 4U5-19 Muons & Eidence for & LENGTH CONTRACTION (continued...) decay of unstable elementary particles called muons demonstrates how length contraction and time dilation complement each other according to a stationary obserer time slows down for muons and they are able to trael further than predicted using classical mechanics (4800 m s 660 m) From a muon s point of iew, time does not change but the distance they trael shortens. December 16, 01 4U5-0 7
U Check Your Learning TEXTBOOK P.591 Q.1-3 December 16, 01 4U5-1 8