In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length−contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than 1g. Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity. (a) Though traveler Anna accelerates, Bob, being on near−inertial Earth, is a reliable observer and will see less time go by on Anna’s clock ( d t ' ) than on his own ( d t ) . Thus, d t ' = ( 1 / γ u ) d t , where u isAnna’s instantaneous speed relative to Bob, Using the result of Exercise 117(c), with g replacing F / m , substitute for u , then integrate to show that t = c g sinh g t ' c (b) How much time goes by for observes on Earth as they “see” Anna age 20 years? (c) Using the result of Exercise 119, show that when Anna has aged a time t ' . She is a distance fromEarth (according to Earth observers) of x = c 2 g ( cosh g t ' c − 1 ) (d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20, how far away from Earth will she end up, and how much time will have passed on Earth?
In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length−contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than 1g. Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity. (a) Though traveler Anna accelerates, Bob, being on near−inertial Earth, is a reliable observer and will see less time go by on Anna’s clock ( d t ' ) than on his own ( d t ) . Thus, d t ' = ( 1 / γ u ) d t , where u isAnna’s instantaneous speed relative to Bob, Using the result of Exercise 117(c), with g replacing F / m , substitute for u , then integrate to show that t = c g sinh g t ' c (b) How much time goes by for observes on Earth as they “see” Anna age 20 years? (c) Using the result of Exercise 119, show that when Anna has aged a time t ' . She is a distance fromEarth (according to Earth observers) of x = c 2 g ( cosh g t ' c − 1 ) (d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20, how far away from Earth will she end up, and how much time will have passed on Earth?
In Example 2.5, we noted that Anna could go wherever she wished in as little time as desired by going fast enough to length−contract the distance to an arbitrarily small value. This overlooks a physiological limitation. Accelerations greater than about 30g are fatal, and there are serious concerns about the effects of prolonged accelerations greater than 1g. Here we see how far a person could go under a constant acceleration of 1g, producing a comfortable artificial gravity. (a) Though traveler Anna accelerates, Bob, being on near−inertial Earth, is a reliable observer and will see less time go by on Anna’s clock
(
d
t
'
)
than on his own
(
d
t
)
. Thus,
d
t
'
=
(
1
/
γ
u
)
d
t
, where u isAnna’s instantaneous speed relative to Bob, Using the result of Exercise 117(c), with g replacing
F
/
m
, substitute for u, then integrate to show that
t
=
c
g
sinh
g
t
'
c
(b) How much time goes by for observes on Earth as they “see” Anna age 20 years? (c) Using the result of Exercise 119, show that when Anna has aged a time
t
'
. She is a distance fromEarth (according to Earth observers) of
x
=
c
2
g
(
cosh
g
t
'
c
−
1
)
(d) If Anna accelerates away from Earth while aging 20 years and then slows to a stop while aging another 20, how far away from Earth will she end up, and how much time will have passed on Earth?
You fly 5000 km across the United States on an airliner at 250 m/s. You return two days later at the same speed. Have you aged more or less than your friends at home? By how much?
Suppose that construction crews have set up a pair of blinking lights 60 m apart to mark a construction zone along a road. A car moves along the road with a speed of 30m/s toward the lights. Let us take the road to be the Home Frame and the car to be the Other Frame. In the Home Frame, the light farther from the car (light 2) blinks 0.66 s before the other light (light 1) blinks as the car approaches. (a) What is the time interval (t'2 - t'1) between the blink events in the car's frame? (Hint: Think about the sign). (b) What is the x displacement x'2 - x'1 between the events in the car frame?
Consider a dragster that, when the light turns green, accelerates uniformly from rest and completes a 372 meter race in a time of 6.61 seconds.
a) During this period, what is the dragster's acceleration expressed as a multiple of the acceleration due to gravity, g?
b) If the dragster could continue with this average acceleration, what would its speed be, in miles per hour, after it has traveled a total distance of 1.6km1.6km, which is approximately one mile?
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