Length contraction, time dilation and relativistic energy

Length contraction is given by $L = \frac{L_0}{\gamma}$. A spacecraft measures $L_0 = 100\ \text{m}$ at rest and travels at $\gamma = 2$. What length does an observer on Earth measure?

The Lorentz factor is $\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$, from which $v = c \times \sqrt{1 - \frac{1}{\gamma^2}}$.

1. a) Calculate the speed of a spacecraft travelling with $\gamma = 2$, expressed as a fraction of $c$.
2. b) Express that speed in $\text{m/s}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$).

A space agency is considering a spacecraft with a proper length of $50\ \text{m}$ measuring only $10\ \text{m}$ from Earth.

1. a) What factor $\gamma$ is needed for this ($L = \frac{L_0}{\gamma}$)?
2. b) Calculate the corresponding speed using $v = c \times \sqrt{1 - \frac{1}{\gamma^2}}$ and express it as a fraction of $c$.
3. c) With this factor $\gamma$, how many years would pass on Earth for each year the crew ages ($\Delta t = \gamma \times \Delta t_0$)?
4. d) In light of the results, evaluate the advantages and problems that interstellar travel at these speeds would present for human exploration.

A phenomenon lasts $\Delta t_0 = 10\ \text{s}$ as measured in its proper reference frame. For an observer seeing it pass at high speed, the Lorentz factor is $\gamma = 2$. Given that time dilation is $\Delta t = \gamma \times \Delta t_0$, how long does the phenomenon last for that observer?

An astronaut travels in a spacecraft at $\gamma = 1.25$. From Earth, the trip is measured to last $8$ years.

1. a) Using $\Delta t = \gamma \times \Delta t_0$, by how many years does the astronaut age?
2. b) If he left at $30$ years of age, how old is he upon return, and how old is his twin brother, who stayed on Earth?

A muon is created in the upper atmosphere travelling at $v = 0.998c$ with $\gamma = 15$; its proper mean lifetime is $2.2\ \mu\text{s}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$).

1. a) Calculate its mean lifetime measured from Earth ($\Delta t = \gamma \times \Delta t_0$).
2. b) Calculate the average distance it travels according to an observer on Earth.
3. c) If the muon had been created at an altitude of $30\ \text{km}$, would it reach the surface on average? Justify your answer using the previous result.

Length contraction only affects the dimension parallel to motion. A spacecraft with a proper length of $50\ \text{m}$ and a proper height of $20\ \text{m}$ travels horizontally at $\gamma = 1.25$. The diagram shows its proper dimensions.

1. a) What length is measured from Earth ($L = \frac{L_0}{\gamma}$)?
2. b) What height is measured from Earth? Justify your answer.

A spacecraft of proper length $L_0 = 120\ \text{m}$ travels at $\gamma = 1.5$.

1. a) What length does an observer on Earth measure ($L = \frac{L_0}{\gamma}$)?
2. b) If the trip lasts $10$ years for Earth, how long does it last for the crew ($\Delta t = \gamma \times \Delta t_0$)?
3. c) Reason which of the two observers measures the proper time of the journey and why.

A spacecraft with a proper length of $100\ \text{m}$ passes through a space station at $\gamma = 7.09$. The station has a hangar of $15\ \text{m}$. The diagram represents the situation.

1. a) What length does the station staff measure for the spacecraft ($L = \frac{L_0}{\gamma}$)? According to them, does the entire spacecraft fit inside the hangar, even if only for an instant?
2. b) For the pilot, it is the station that moves: what length does the pilot measure for the hangar? Does the spacecraft fit according to him?
3. c) The apparent contradiction is resolved because "the spacecraft is entirely inside at the same time" does not mean the same thing to both observers. Name and explain the relativistic concept that resolves the paradox.

The energy associated with a rest mass is $E = m \times c^2$, with $c = 3 \times 10^{8}\ \text{m/s}$. Calculate the energy corresponding to $1\ \text{kg}$ of mass and express it in standard form.

The Sun radiates a power of $3.9 \times 10^{26}\ \text{W}$ by converting part of its mass into energy.

1. a) Calculate the energy it emits each second.
2. b) Using $E = m \times c^2$ with $c = 3 \times 10^{8}\ \text{m/s}$, calculate the mass that the Sun converts into energy each second.
3. c) Express the result in tonnes ($1\ \text{t} = 10^{3}\ \text{kg}$).

In a positron emission tomography (PET) scan, a positron emitted by a radioactive tracer annihilates with an electron in the tissue, and all its mass is converted into energy, emitted as two gamma photons. The mass of each particle is $9.11 \times 10^{-31}\ \text{kg}$ (given: $c = 3 \times 10^{8}\ \text{m/s}$). The diagram represents the process.

1. a) Calculate the total mass annihilated in the event.
2. b) Calculate the energy released ($E = m \times c^2$).
3. c) What energy does each photon carry if the two share the energy equally?
4. d) Explain why matter-antimatter annihilation is the most efficient possible process for converting mass into energy, and compare it with solar fusion, which converts less than $1\ \%$ of the mass.