The magnitude of the velocity of the Cyclist A relative to the Cyclist B is

  Observations of muon decay can be used as evidence of

I. Time Dilation
II. Length Contraction
III. Absolute Simultaneity

  a. A muon is created in a high-energy cosmic ray interaction in Earth’s uppen4b driq w z+w-7p3hj(r atmjw 3bq7zp4i dr(hn-+w osphere. The lifetime of the muon before decay is $2.3 \times 10^{-6} \, s$ in its frame of reference. The lifetime of the muon is measured as $11 \times 10^{-6} \, s$ from the perspective of an observer on Earth.

i. State and explain which of the given lifetimes is the proper time.
ii. Calculate, according to the theory of special relativity, the speed of the muon is around    % of the speed of light.
iii. Muons are likely created at $10 \, km$ above the earth’s surface. Most of the muons are detected on the surface of the Earth. Explain how this observation can be interpreted as proof of special relativity.

b. A space-time diagram shows a motion of a muon in the frame of reference of a stationary observer at the place where the muon is created.

i. By using the information in a.(ii), calculate the angle $θ$ shown on the graph. $\theta$ =    $^{\circ}$
ii. Another muon is created a few kilometres below the muon in b.(i) at the same time. It is moving at $0.75$c. Draw the worldline of the second muon on the graph in b.(i) according to the same observer.

  a. In a thought experiment, a spacecraft recedes away fro9a) iy48vm7rzh bta4: xg t3lim the Earth at a constant speed of 0.92c in the reference frame of Earth. 497tl8ait34rmv azxyib) gh: The length of the spacecraft is measured as 35m by an observer in the spacecraft.



i. State the differences between Galilean and Special relativity with regards to the speed of light and time.
ii. Calculate the length of the spacecraft measured by an observer on Earth.    $m$

b. After some time, the spacecraft encounters a rocket that moves at a speed of 0.75c relative to Earth in the opposite direction.
i. Calculate, relative to the spacecraft, the speed of the rocket according to the Galilean transformation is around    times the speed of light.
ii. Calculate, relative to the spacecraft, the speed of the rocket according to the Lorentz transformation. v =    % the speed of light.
iii. According to postulates of special relativity, outline which of your calculations in (b)(i) and (b)(ii) related to the speed of the rocket relative to the spacecraft is valid.

  

  At a relativistic speed, a rocket journeyed from Earth to Saturn. The durati0ydi so3f6 nj -2kihnu8zup( /on o2f6un 0hpdk- 8/nu3izy ios(jf the travel is measured according to the stationary observer on earth as $t$. The length of the rocket is measured according to the observer in the rocket as $L$. Which of the following is correct for the time of travel according to the observer in the rocket and the length of the rocket according to the observer on the Earth?

  A spaceship is moving at a constant velocity relative to a station. The spacetime diagram shows the worldline of the spaceship in the frame of reference of the station.


Which of the following gives the speed of the spaceship in terms of $c$?

  In a distant future, an av/ y4 /caoh5iprea with dimensions of $2.0ly$ and $5.0ly$, as measured by a stationary observer on Earth, is reserved for a spaceship road. The diagram illustrates the area and the direction of motion for a spaceship moving at a speed of $0.6c$



What is the area of the road from the perspective of the observer in the spaceship?

  1.Car $A$ and Car $B$ are travelling in the same direction along a straight line. In the reference frame of a stationary observer on the ground, Car $A$ has a speed of $24ms^-1$ and Car B has a speed of $36ms^-1$.


1.Explain what is meant by a frame of reference.
2.Calculate the velocity of Car $B$ relative to the stationary observer in Car $A$. $u'$ =   $ms^-1$
3.The distance travelled by Car $B$ relative to Car $A$ is $60m$ in a time interval of $5.0s$. Calculate the distance travelled by Car $B$ relative to the stationary observer on the ground for the same time interval. $Δx$ =   $m$


2.Now consider two spaceships, $C$ and $D$, traveling through space towards a planet.



Both of the spaceships turn on their lights. The turning on of the lights of spaceship $C$ and the turning of the lights of spaceship $D$ are defined as Events $C$ and $D$ respectively. Event C has coordinates $x_C=3.456\times10^{5} $,$t_C=0s$ and event $D$ has coordinates $x_D=3.443\times10^{5}$, $t_D=1.1\times10^{-3}s$ according to a stationary observer on the planet.
A rocket flies towards the two spaceships at a speed of $0.85c$ according to the same observer.
1.According to the frame of reference of the rocket, determine the space differences between events $C$ and $D$. $Δx′$ =    $\times10^{-5}\,m$
2.According to the frame of reference of the rocket, determine the space-time interval between events $C$ and $D$. $(Δs′)^2$ =   $\times10^{11}\,m^2$
3.Discuss whether Event $C$ and Event $D$ can occur at the same time according to some other inertial frame. $(Δs)^2$ =   

  A spaceship is moving at a speedkqr v)oaxhi ,e 7b/53znah4x4 of $0.8c$ relative to a stationary observer on Earth. An event occurring in the spaceship is measured by an observer in the spaceship to have a duration of $12s$. What is the duration of the event as measured by the stationary observer on Earth?

  In a landing zone designed for spaceships that can move at relativistic speeds, a sequence of warning lights A, B, C, and D are used to guide landings. The spacetime diagram of this system shows the lights at rest in a particular reference frame and the worldline of a spaceship approaching the landing zone.


Which of the following is the order of the lights reaching an observer on the spaceship from warning lights?