(a) In communicating with an astronaut on the moon, 3.8 x 108 m from earth, what is the minimum time delay in getting a response to a question? (b) What would the minimum time delay be for communication with an astronaut at a distance of 2 light-years from earth?
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- Two astronomical events are observed to occur at a time of 0.30 s apart and a distance separation of 2.0109m from each other. How fast must a spacecraft travel from the site of one event toward the other to make the events occur at the same time when measured in the frame of reference of the spacecraft?Check Your Understanding Suppose a space probe moves away from Earth at a speed 0.350c. It sends a radio-wave message back to Earth at a frequency of 1.50 GHz. At what frequency is the message received on Earth?Suppose the primed and laboratory observers want to measure the length of a rod that rests on the ground horizontally in the space between the helicopter and the tower (Fig. 39.8B). To derive the length transformation L = L (Eq. 39.5), we had to assume that the positions of the two ends were determined simultaneously. What happens to the length transformation equation if both observers measure the end below the helicopter at one time t1 and the other end at a later time t2?
- An Earth satellite used in the Global Positioning System moves in a circular orbit with period 11 h 58 min. (a) Determine the radius of its orbit. (b) Determine its speed. (c) The satellite contains an oscillator producing the principal nonmilitary GPS signal. Its frequency is 1 575.42 MHz in the reference frame of the satellite. When it is received on the Earths surface, what is the fractional change in this frequency due to time dilation, as described by special relativity? (d) The gravitational blueshift of the frequency according to general relativity is a separate effect. The magnitude of that fractional change is given by ff=Ugmc2 where Ug/m is the change in gravitational potential energy per unit mass between the two points at which the signal is observed. Calculate this fractional change in frequency. (e) What is the overall fractional change in frequency? Superposed on both of these relativistic effects is a Doppler shift that is generally much larger. It can be a redshift or a blueshift, depending on the motion of a particular satellite relative to a GPS receiver (Fig. P1.39).Check Your Understanding a. A particle travels at 1.90108m/sand lives 2.10108swhen at rest relative to an observer. How long does the particle live as viewed in the laboratory? b. Space craft A and B pass in opposite directions at a relative speed of 4.00107m/s . An internal clock in space craft A causes it to emit a radio signal for 1.00 s. The computer in spacecraft B corrects for the beginning and end of the signal having traveled different distances, to calculate the time interval during which ship A was emitting the signal. What is the time interval that the computer in spacecraft B calculates?Show that (x,t)=Asin(kxt) and (x,t)=Acos(kxt) do not obey Schrödinger's time-dependent equation.
- (a) In communicating with an astronaut on the moon, 3.8 108 m from earth, what is the minimum time delay in getting a response to a question?(b) What would the minimum time delay be for communication with an astronaut at a distance of 6 light-years from earth?(a) How long in seconds does it take a radio signal to travel 180 km from a transmitter to a receiving antenna? (b) We see a full Moon by reflected sunlight. How much earlier did the light that enters our eye leave the Sun? The Earth – Moon and Earth – Sun distances are 3.8x105 km and 1.5 × 108 km, respectively. (c) What is the round-trip travel time in seconds for light between Earth and a spaceship at a 6.8 × 107 km distance from Earth? (d) Suppose astronomers observe a supernova about 7300 light-years (ly) distant. How long ago in years did the explosion actually occur?The Voyager 1 spacecraft has traveled farther than any other man-made object, and inAugust 2012 it entered into interstellar space when it was a distanceof 1.8 * 1013 m from Earth. How many hours elapsed between thetime a command was sent from Earth and the time the commandwas received by Voyager when it entered interstellar space?
- (a) How long does it take a radio signal to travel 150 km from a transmitter to a receiving antenna? (b) We see a full Moon by reflected sunlight. How much earlier did the light that enters our eye leave the Sun? The Earth–Moon and Earth–Sun distances are 3.8 *10^5 km and 1.5* 10^8 km, respectively. (c) What is the round-trip travel time for light between Earth and a spaceship orbiting Saturn, 1.3*10^9 km distant? (d) The Crab nebula, which is about 6500 light-years (ly) distant, is thought to be the result of a supernova explosion recorded by Chinese astronomers in A.D. 1054. In approximately what year did the explosion actually occur? (When we look into the night sky, we are effectively looking back in time.)The cosmic rays of highest energy are protons that have kinetic energy on the order of 1013 MeV. (a) From the point of view of the proton, how many kilometers across is the galaxy? (b) How long would it take a proton of this energy to travel across the Milky Way galaxy, having a diameter ~ 105 light-years, as measured in the proton’s frame?Muons are unstable subatomic particles that decay to electrons with a mean lifetime of 2.2 µs. They are produced when cosmic rays bombard the upper atmosphere about 10 km above the earth’s surface, and they travel very close to the speed of light. The problem we want to address is why we see any of them at the earth’s surface. (a) What is the greatest distance a muon could travel during its 2.2 µs lifetime? (b) According to your answer in part (a), it would seem that muons could never make it to the ground. But the 2.2 µs lifetime is measured in the frame of the muon, and muons are moving very fast. At a speed of 0.999c, what is the mean lifetime of a muon as measured by an observer at rest on the earth? How far would the muon travel in this time? Does this result explain why we find muons in cosmic rays? (c) From the point of view of the muon, it still lives for only 2.2 µs, so how does it make it to the ground? What is the thickness of the 10 km of atmosphere through which the muon…