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TN2612 - introduction to special relativity Lambert van Eijck NPM2, dep. Radiation Science and Technology, fac. Appl. Sci, TU Delft 2018-2019 Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 1 Course info Teacher: Lambert van Eijck, l.vaneijck@tudelft.nl Lectures: 1+8 colleges, 1+8 werkcolleges Book: Dynamics and Relativity, Ch 11 t/m 14 Author: W.D. Comb Publisher: Oxford Examination: written Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 2 today: classical mechanics as context for special relativity Figure: Galileo Galilei (1564-1642) in Pisa: which object falls faster: the light or the heavy object? Defines speed, acceleration, gravitational acceleration. Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 3 a typical success story of classical mechanics: dynamics of many particles Figure: i particles with mass mi , position xi , yi , zi , velocity v~i . Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 4 dynamics of many particles Figure: these i particles can be stars, planets, steel balls or atoms! Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 5 the most famous person in classical mechanics is Isaac Newton (1642-1727) ~ = m · ~a F In the case of hard spheres, the trajectories of the individual particles are described as: ~xi (t) = ~xi0 + ~vi0 t + 0.5~ai t 2 and the ~ = m · ~a interactions are described using: F Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 6 center of mass The center of mass of a system of particles: ~xi (t) = ~xi0 + ~vi0 t + 0.5~ai t 2 is defined as: ~ cm = X P xi i m·~ P i m Lambert van Eijck The center of mass is a position defined relative to an object or system of objects. It is the average position of all the parts of the system, weighted according to their masses. For simple rigid objects with uniform density, the center of mass is located at the centroid. there should be an i subscript for m too!! TN2612 - introduction to special relativity - lecture 0 7 velocity of the center of mass ~ cm = Similarly, the velocity vectors ~vi can be added: V Lambert van Eijck P vi i mi ·~ P i mi TN2612 - introduction to special relativity - lecture 0 8 defining an inertial frame of reference: ~ cm (t) = X ~ cm (0) + V ~ cm · t X position of center of mass at time t is its initial position (t=0) + its speed times time passed the second derivative of displacement x is acceleration a P m ·~ẍ The acceleration of the center of mass aCM is: ~acm = Pi mi i i We i imply that there are no external forces and: the sum of all m for i is equal to the total P ~ mass of the system i Ẍcm = 0 the total acceleration of the center of mass is 0 Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 9 inertial frame of reference for 2 particles: ~ cm (t) = X ~ cm (0) + V ~ cm · t X Figure: Two particles (red) trajectory can be ”summarized” as one CM system (green). For the two particles system i, j: mi ~ẍi + mj ~ẍj = 0 Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 10 relative velocities in two reference frames that's because you move with it, so there is no relative motion between you and it If you decide to ’sit’ on the CM reference frame, then VCM = 0 for you. The relative velocities ∆v21 of the (red) particles 1 and 2 0 , according to me should be the same as the relative velocity v21 according to you. ~v21 = 0 ~v21 Lambert van Eijck = ∆~x2 ∆t ∆~x20 ∆t 0 − − ∆~x1 ∆t ∆~x10 ∆t 0 relative velocity is described as the difference in velocity between one particle (Dx2/t) and the other (Dx1/t) for a different observer, these relative velocities are going to be different, but the difference between them should be the same TN2612 - introduction to special relativity - lecture 0 11 relative velocities in two reference frames ~v21 = 0 = ~v21 ∆~x2 ∆t ∆~x20 ∆t 0 − − ∆~x1 ∆t ∆~x10 ∆t 0 so, ~v21 = 0 = ~v21 Lambert van Eijck (x2 −x1 )tB −(x2 −x1 )tA tB −tA (x20 −x10 )t 0 −(x20 −x10 )t 0 B tB0 −tA0 A TN2612 - introduction to special relativity - lecture 0 12 relative velocities in two reference frames ~v21 = 0 = ~v21 (x2 −x1 )tB −(x2 −x1 )tA tB −tA (x20 −x10 )t 0 −(x20 −x10 )t 0 B tB0 −tA0 A Although positions x might be unequal for us (x 6= x 0 ), in general, any distance (= relative positions!) ∆x = ∆x 0 is the same for you and for me. This, however, implies that we have determined these distances at tA = tA0 and tB = tB0 and so ∆t = ∆t 0 . Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 13 ~ = m · ~a F Figure: Classical mechanics is based on standards of length and time (which we will have to sacrifies for Special Relativity). ~ = m · ~a are so successful that we (you) Classical mechanics and F use it still today, on a daily basis. It is based on the notion of distance and time intervals that seem ’set in stone’. Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 14 inertial frame of reference for 2 particles: ~ cm (t) = X ~ cm (0) + V ~ cm · t X Figure: The collision of two particles can be described in the CM system. How to describe the collision of the two particles system 1, 2 with mass m (Christiaan Huygens)? Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 15 ~ = m · ~a F I I I I I describe the system in an inertial frame of reference ~ is needed to change the motion of mass m force F ~ and ~a are vectors, implying addition F P ~ P ~ a or equivalently: so rather: i Fi = m · ~ i Fi = d~p dt momentum: p = m*v F = dp/dt intuitively, force causes acceleration, but does the formula imply such causality? Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 16 Exercise 1 Figure: Wikipedia: ”The froghopper can accelerate at 4000 m/s2 over 2 mm as it jumps” I I I what force is directly responsible for the acceleration of the froghopper? How big is this force if the acceleration is vertical and the insect weighs 0.1g? What is the height that the froghopper reaches? Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 17 Exercise 2 Usain Bolt gives you a ’discount’ of 20m on a 100m running competition (you start at x = 20m in front of him). I What average speed do you need to prevent him from winning (assuming he is as fit as ever)? You lost anyway, so Usain offers you to cross the starting position x = 0 at your full speed. I What average speed do you need to run, to win? I What is the distance you’ve run, according to you? I What time did it take you to win this sprint, according to your watch? Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 18 force and energy ”before” the era of E = mc 2 R ~ · dx ~ work done by a force is defined as: W = F 2 kinetic energy is defined as: ∆K = m · v /2 potential energy is defined as: U = k · (∆x)2 /2 or U = mgh, or ... Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 19 Exercise 3 An intergalactic rocket with kinetic energy K explodes into two pieces, each of which moves with twice the speed of the original object. I determine the change in direction of each of the pieces w.r.t. the flight direction prior to the explosion. I calculate the ratio of the internal and center-of-mass energies int . after the explosion KKCM I which conservation laws did you use for your calculations? I which conservation laws did you not use for your calculations? Lambert van Eijck TN2612 - introduction to special relativity - lecture 0 20