Concept #1: Time Dilation

Example #1: Time Dilation for a Muon from the Atmosphere

Practice: The international space station travels in orbit at a speed of 7.67 km/s. If an astronaut and his brother start a stop watch at the same time, on Earth, and then the astronaut spends 6 months on the space station, what is the difference in time on their stopwatches when the astronaut returns to Earth? Note that 6 months is about 1.577 x 10^{7} s, and c = 3 x 10 ^{8} m/s.

Concept #2: Length Contraction

Example #2: Length Contraction for a Muon from the Atmosphere

Practice: In the following figure, a right triangle is shown in its rest frame, S'. In the lab frame, S, the triangle moves with a speed v. How fast must the triangle move in the lab frame so that it becomes an isosceles triangle?

Concept #3: Proper Frames and Measurements

Muons are commonly produced in the upper atmosphere due to collisions of cosmic rays with the atoms in the atmosphere. The muons produced commonly travel downwards at speeds up to 0.99 c. If the half life of the muon, in the rest frame, is 2.2 μs,
(a) How long is the muon's half life, according to an observer at rest on the Earth's surface
(b) How far can the muon travel, according to an observer at rest on the Earth's surface?

Ted travels in a spaceship at 0.45 c relative to Alice, who's at rest on the Earth's surface. Ted travels for 20s, as measured on his watch.
(a) Who measures the proper time, Ted or Alice?
(b) How much time elapses on Alice's watch during this motion?

An astronaut travels at 0.8 c from Earth to a star that is 20 light years away from Earth, as measured from the Earth's perspective. If the astronaut is 25 years old when leaving Earth, what is the age of the astronaut when arriving at the star?

A spaceship passes an Earth-bound observer at a speed v. The observer on Earth measures the ship to be 300 m long, but an astronaut on the ship would measure it to be 400 m long. What is the speed of the ship, v?

A spaceship passes an observer on Earth at a speed of 0.7 c. On the ship, an antenna has a length of 1 m and an angle of 15o measured from the surface of the ship. When passing the Earth-bound observer, what length and angle is the antenna observed to have?

A spaceship flies at a speed of 0.4c towards a stationary mirror. The ship sends a laserbeam at a wavelength of 500 nm towards the mirror, which reflects back towards the ship. What wavelength of light does the ship measure coming off of the mirror?

A subatomic particle called the pion has a measured lifetime of 4.4x10-8 s. If the lifetime of a pion measured at rest is 2.6x10-8 s, how fast is the pion moving in the lab frame?

At what speed does time dilation have a "significant" effect? Assume that significant means a difference in 1% of dilated time and proper time.

Manufactured on Earth, a cube has a side length of 2.0 cm. As a spaceship flies past an observer on Earth, it fires the cube at a speed of 0.7 c relative to the observer on Earth. What volume for the cube would the observer on Earth measure?

In The Planet of the Apes, humans "traveled" into the future by hibernating on a spaceship traveling relative to the Earth for 120 years before returning. If the ship traveled at 99.99% the speed of light, how much time would have elapsed on Earth while they hibernated?

A flight in an airplane going 300 m/s takes one hour as measured by a person in the airplane. How long did this flight take according to an observer on the ground?A. 1.8 ns less than 1.0 hourB. 3.6 ns less than 1.0 hourC. Exactly 1.0 hour.D. 3.6 ns more than 1.0 hourE. 1.8 ns more than 1.0 hour

A flight in a space ship going half the speed of light takes one hour as measured by a person in the airplane. How long did this flight take according to an observer on the ground?A. 4.00 hrB. 1.15 hrC. Exactly 1.0 hour.D. 0.500 hrE. 0.563 hr

Anna boards a spaceship and does an interstellar flight while her twin brother, Carlos waits on Earth. Anna’s ship travels at β = 0.80 and, according to Carlos, is gone for 10 years. According to Anna, how long was she gone?A. 3.0 yearsB. 6.0 yearsC. 10. yearsD. 16 yearsE. 19 years

An arrow of length 1.00 m flies by you at 0.8c. You observe the length of the arrow to beA. shorter than 1 mB. 1 m in lengthC. longer than 1 m

An arrow of length 1.00 m flies by you at 0.8c. If you were to measure the time it takes to fly past you, what would you measure this to be?A. 0.75 nsB. 1.25 nsC. 2.5 nsD. 4.2 nsE. 6.9 ns

An arrow of length 1.00 m flies by you at 0.8c. In the reference frame of the arrow, how long does it take to pass by you?A. 0.75 nsB. 1.25 nsC. 2.5 nsD. 4.2 nsE. 6.9 ns

Samir (who is standing at rest on the ground) starts his stopwatch at the instant that Maria flies past him in her spaceship. She continues to fly her spaceship at the same constant velocity. According to Maria, at the instant that Samir’s stopwatch reads 8.0 s, Maria’s stopwatch reads 10.0 s. According to Maria, her spaceship is 100 m long (along the direction of motion). According to Samir, the length of Maria’s spaceship isA. 64 m.B. 80 m.C. 100 m.D. 125 m.E. Not enough information is given to decide.

Red light is emitted from a source at rest at a wavelength of 650 nm. How fast would this source have to be moving relative to you for you to see it as blue? Would it need to be moving towards or away from you?

It is known that the universe is expanding, so all stars are moving away from us, and the further away they are, the faster they are moving away from us. For convenience, there is a quantity known as redshift, z, that can describe how "far" an object is away from us. Redshift is given by the equation
1 + z = λ'/λo
where λo is the wavelength of light emitted at rest and λ' is the wavelength of light observered on Earth.
(a) Is redshift a distance measurement?
(b) If a star is supposed to emit light at a frequency of 4.57x1014 Hz, what would the observed frequency be if the redshift of this star were 0.1?