Foundations of Modern Physics Homework#1 Due by: 2015.04.07. (Tue) 12:30 #1. Show that if length is contracted by the factor �1 − 𝑣2 𝑐2 in the direction of motion, then the result in Equation (2.3) ∆𝑡 = 𝑡2 − 𝑡1 will have the factor needed to make ∆𝑡 = 0 as needed by Michelson and Morley. Assume that ℓ1 = ℓ2 . #2. Two events occur in an inertial system K as follows: Event1: 𝑥1 = 𝑎, 𝑡1 = 2𝑎 𝑐 Event2: 𝑥2 = 2𝑎, 𝑡2 = , 𝑦1 = 0, 𝑧1 = 0 3𝑎 2𝑐 , 𝑦2 = 0, 𝑧2 = 0 In what frame K’ will these events appear to occur at the same time? Describe the motion of system K’. #3. Show that the experiment depicted in Figure 2.11 and discussed in the text leads directly to the derivation of length contraction. #4. A mechanism on Earth used to shoot down geosynchronous satellites that house laser-based weapons is finally perfected and propels golf balls at 0.94c. (Geosynchronous satellites are placed 3.58 × 104 𝑘𝑘 above the surface of the Earth.) (a) What is the distance from the Earth to the satellite, as measured by a detector placed inside the golf ball? (b) How much time will it take the golf ball to make the journey to the satellite in the Earth’s frame? How much time will it take in the golf ball’s frame? #5. Three galaxies are aligned along an axis in the order A, B, C. An observer in galaxy B is in the middle and observes that galaxies A and C are moving in opposite directions away from him, both with speeds 0.60c. What is the speed of galaxies B and C as observed by someone in galaxy A? #6. Prove that for a spacelike interval, two events cannot occur at the same place in space. #7. Given two events, (𝑥1 , 𝑡1 ) and (𝑥2 , 𝑡2 ), use a spacetime diagram to find the speed of a frame of reference in which the two events occur simultaneously. What values may ∆𝑠 2 have in this case? 𝑑𝑝⃗ #8. Newton’s second law is given by 𝐹⃗ = . If the force is always perpendicular to the velocity, show that 𝐹⃗ = 𝑚𝑚𝑎⃗, where 𝑎⃗ is the acceleration. 𝑑𝑑 #9. Show that linear momentum is conserved in Example 2.9 as measured by Mary. #10. Calculate the momentum, kinetic energy, and total energy of an electron traveling at a speed of (a) 0.020c (b) 0.20c (c) 0.90c.