Lorentz Transformations
A spaceship is moving at 0.6c relative to Earth. An observer on the spaceship measures the length of a rod to be 3 meters. Calculate the length of the rod as measured by an observer on Earth.
gamma = 1.25
rest length = gamma * relativistic length
rest length = 1.25 * 3
rest length = 3.75 m
You are observing a nearby star from Earth. Over a six-month period, you measure its apparent shift in position against a distant background star. The angle of parallax is 0.5 arcseconds. Calculate the distance to the nearby star.
d = 1 / p
d = 1 / 0.5
d = 2 parsecs
A moon orbits a distant gas giant with a semi-major axis of 500,000 kilometers and has an orbital period of 20 days. Calculate the mass of the gas giant.
M = a^3 / T^2
M = 500000^3 / 20^2
M = 3.125 ∙ 10^14 kg
If a clock on a spaceship traveling at 0.8c reads 10 minutes, what is the time elapsed on Earth for the same event?
gamma = 1.67
relativistic time = rest time * gamma
relativistic time = 10 * 1.67
relativistic time = 16.7 min
A particle is moving at 0.9c. Determine its relativistic mass if its rest mass is 2 kg.
gamma = 2.29
relativistic mass = gamma * rest mass
relativistic mass = 2.29 * 2
relativistic mass = 4.58 kg
The Hubble Space Telescope observes a distant galaxy and notices an apparent shift in its position against a more distant galaxy. The angle of parallax is 0.002 arcseconds. Calculate the distance to the observed galaxy.
d = 1 / p
d = 1 / 0.002
d = 500 parsecs
A distant asteroid orbits the Sun with a semi-major axis of 2.5 astronomical units (AU) and has an orbital period of 5 years. Calculate the mass of the asteroid.
M = a^3 / T^2
M = 2.5^3 / 5^2
M = 0.625 kg
Calculate the Lorentz factor for an object moving at 0.5c.
gamma = 1.15
A spaceship is 40 meters long when at rest. What is its length when moving at 0.9c relative to Earth?
gamma = 2.29
relativistic length = rest length / gamma
relativistic length = 40 / 2.29
relativistic length = 17.47 m
Astronomers use a ground-based telescope to observe an asteroid as it passes relatively close to Earth. The observed angle of parallax is 20 arcseconds. Calculate the distance to the asteroid.
d = 1 / p
d = 1 / 20
d = 0.05 parsecs
A satellite orbits a distant exoplanet with a semi-major axis of 500,000 kilometers and has an orbital period of 5 hours. Calculate the mass of the exoplanet.
M = a^3 / T^2
M = 500000^3 / 5^2
M = 5 ∙ 10^15 kg
A train is moving at 0.85c, and its length is 100 meters. What is its contracted length as observed from a stationary observer?
gamma = 1.9
relativistic length = rest length / gamma
relativistic length = 100 / 1.9
relativistic length = 52.68 m
A particle is moving at 0.99c. Determine its relativistic mass if its rest mass is 1 gram.
gamma = 7.09
relativistic mass = gamma * rest mass
relativistic mass = 7.09 * 1
relativistic mass = 7.09 g
You observe a nearby exoplanet in a distant star system from Earth. Over a period of time, you measure its apparent shift against its host star. The distance is 25 parsecs. Calculate the angle of parallax of the exoplanet.
d = 1 / p
p = 1 / d
p = 1 /25
p = 0.04 arcsec
A moon orbits a distant ice giant with a mass of 5 ∙ 10^15 kg has an orbital period of 40 days. Calculate the semi-major axis of the ice giant in km.
M = a^3 / T^2
a^3 = M ∙ T^2
a^3 = (5 ∙ 10^15) ∙ 40^2
a = cubed root [ (5 ∙ 10^15) ∙ 40^2 ]
a =2,000,000 km
A rod is 2 meters long in its rest frame. Calculate its length as observed by an observer moving at 0.8c relative to the rod.
gamma = 1.67
relativistic length = rest length / gamma
relativistic length = 2 / 1.67
relativistic length = 1.2 m
An observer on Earth measures the length of a spaceship to be 40 meters, but an observer on the spaceship measures it to be 32 meters. Calculate the speed of the spaceship.
relativistic length = rest length / gamma
32 = 40 / gamma
gamma = 1.25
Therefore v = 0.6c
You observe a binary star system from Earth. The two stars have a significant apparent separation, and you measure the distance between them to be 2 parsecs. Calculate the parallax angle to the binary star system.
d = 1 / p
p = 1 / d
p = 1 /2
p = 0.5 arcsec
A comet orbits the Sun with a semi-major axis of 3.5 AU has a mass of 0.0686 kg. Calculate the comet's orbital period in years.
M = a^3 / T^2
T^2 = a^3 /M
T = sqrt( a^3 /M )
T = sqrt( 3.5^3 /0.0686 )
T = 25 years
Iron Man invents a god-level armor, with Einstein's help OF COURSE, and takes off to battle Kang the Conqueror. As he flies across the cosmos, Jarvis alerts him that his T_uv tensor has quadrupled! Captain Marvel sees him zoom by her while she travels back to Earth. If Iron Man is normally 6.3 ft tall, how tall will he appear to Captain Marvel?
gamma = 4
relativistic length = rest length / gamma
relativistic length = 6.3 / 4
relativistic length = 1.58 ft