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- The spaceship Lilac, based on the Purple Planet, is779 m long when measured at rest. When the Lilac passes Earth, observers there measure its length to be 724 m. At what speed v is the Lilac moving with respect to Earth? Express your answer as a multiple of the speed of light ?.Observer O fires a light beam in the y direction (vy = c). Use the Lorentz velocity transformation to find v′ x and v′y and show that O′ also measures the value c for the speed of light. Assume O′ moves relative to O with velocity u in the x direction.A car is moving relative to a pedestrian with a Lorentz factor of 1.45. With the length of the car being 5.7m according to the pedestrian, what's the length according to the driver of the car (he's moving at the same speed of the car) ?
- An alarm clock is set to sound in 10.0 = h. Att 0, the clock is placed in a spaceship moving with a speed of 0.736 c (relative to Earth). What distance, as determined by an Earth observer, does the spaceship travel before the alarm clock sounds? (Hint: Keep track of your units!) answer in mAlice and Mary are twins. Mary moves away from the Earth to a distant planet 12 light years away from the Earth. a) With what MINIMUM speed should Mary move in order to return to the Earth N=6 years younger than her sister? b) Explain the resolution of "twin paradox"? ( The twins seem to be in a symmetrical situation, why is one of them younger.)Please derive the attatched velocity transformations by using the Lorentz transformation for the velocity four vector (uμ). (hint: two of the Lorentz transformation equations are needed to solve this)
- At t=0, an alien spaceship passes by the earth: let this be event A. At t=13 min (according to synchronized clocks on earth and Mars), the spaceship passes by Mars, which is 5 light-minutes from earth at the time: let this be event B. Radar tracking indicates that the spaceship moves at a constant velocity between earth and Mars. Just after the ship passes earth, people on earth launch a probe whose purpose is to catch up with and investigate the spaceship. This probe accelerates away from earth, moving slowly at first, but moving faster and faster as time passes, eventually catching up with and passing the alien ship just as it passes Mars. In all parts of this problem, you can ignore the effects of gravity and the relative motion of earth and Mars (which are small) and treat earth and Mars as if they were both at rest in the inertial reference frame of the solar system. Also assume that both the probe and the alien spacecraft carry clocks. 1. Draw a quantitatively accurate…A.8. Show with the aid of the Lorentz transformation (A.28) that the quantity c²₁² — x² - y² – z² is an invariant, namely c²1² − (.x² + ¸‚µ‚² + =²) = (²² 1¹² — (x²² + 1,²² 1₁² +=²²)