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 $v = c \cdot \sqrt{1 - \frac{1}{\gamma^2}}$.

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: $c = 3 \cdot 10^{8}\ \text{m/s}$).

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 $\Delta t = \gamma \cdot \Delta t_0$, 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 $\Delta t = \gamma \cdot \Delta t_0$, 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 at an altitude of $15\ \text{km}$, travels at $v = 0,9995c$ with $\gamma = 31,6$, and its proper mean lifetime is $2,2\ \mu\text{s}$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$). The diagram compares both viewpoints.

1. a) Calculate its mean lifetime seen from Earth ($\Delta t = \gamma \cdot \Delta t_0$).
2. b) Calculate the average distance it travels according to Earth and conclude whether it can reach the surface.
3. c) From the muon's point of view, its lifetime is $2,2\ \mu\text{s}$ but the atmosphere contracts: calculate the thickness of the atmosphere it passes through ($L = \frac{L_0}{\gamma}$) and the distance it travels in its proper lifetime, and check that you reach the same conclusion as in b).
4. d) Explain why both observers agree on the final result (the muon reaches the ground) even though they explain it differently.

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 ($\Delta t = \gamma \cdot \Delta t_0$)?
3. c) Reason which of the two observers measures the proper time of the journey and why.

Proxima Centauri, the closest star to the Sun, is $4,2$ light years away as measured from Earth. A spacecraft travels there at $\gamma = 2$, which corresponds to $v = 0,866c$ (obtained from $v = c \cdot \sqrt{1 - \frac{1}{\gamma^2}}$). Remember that a light year is the distance light travels in one year. The diagram summarizes the journey.

1. a) How long does the journey last for observers on Earth?
2. b) How long does it last for the crew ($\Delta t = \gamma \cdot \Delta t_0$)?
3. c) What Earth–star distance does the crew measure ($L = \frac{L_0}{\gamma}$)?
4. d) Check the consistency of the results: the crew must travel the contracted distance from part c) at speed $0,866c$ in the time from part b). Verify this numerically.

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

The Sun radiates a power of $3,9 \cdot 10^{26}\ \text{W}$ by converting part of its mass into energy.

1. a) Calculate the energy it emits each second.
2. b) Using $E = m \cdot c^2$ with $c = 3 \cdot 10^{8}\ \text{m/s}$, 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 $9,11 \cdot 10^{-31}\ \text{kg}$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$). The diagram represents the process.

1. a) Calculate the total mass annihilated in the event.
2. b) Calculate the energy released ($E = m \cdot c^2$).
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.

The Sun has a mass of $2 \cdot 10^{30}\ \text{kg}$ and radiates a power of $3,9 \cdot 10^{26}\ \text{W}$ by converting mass into energy ($E = m \cdot c^2$; $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the mass the Sun loses each second.
2. b) Calculate the mass lost in $10^{10}$ years ($3,2 \cdot 10^{17}\ \text{s}$), which is of the order of its total lifetime.
3. c) What fraction of the solar mass does this represent? Express it as a percentage.
4. d) Conclude whether radiation significantly changes the Sun's mass over its lifetime and comment on why its gravity is barely affected.