Exercises: Length contraction, time dilation, and relativistic energy

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?

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 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.

A rod has a proper length of $2\ \text{m}$ and moves relative to the laboratory at $\gamma = 1,25$. Using $L = \frac{L_0}{\gamma}$, calculate its length measured from the laboratory.

A muon has a mean lifetime at rest of $2,2\ \mu\text{s}$. A beam of muons travels through a storage ring at $\gamma = 10$. Applying $\Delta t = \gamma \cdot \Delta t_0$, what is the mean lifetime of the muons measured from the laboratory?

Calculate the rest mass energy of an electron, $m = 9,11 \cdot 10^{-31}\ \text{kg}$, using $E = m \cdot c^2$ with $c = 3 \cdot 10^{8}\ \text{m/s}$. Express the result in standard form to two significant figures.

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 clock travels in a spacecraft at $\gamma = 1,5$. When $6$ years have passed for the traveller, how many years have elapsed on Earth? Apply $\Delta t = \gamma \cdot \Delta t_0$ and state which of the two times is the proper time of the trip.

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 in the upper atmosphere travelling at $v = 0,998c$ with $\gamma = 15$; its proper mean lifetime is $2,2\ \mu\text{s}$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate its mean lifetime measured from Earth ($\Delta t = \gamma \cdot \Delta t_0$).
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.

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.

One kilowatt-hour, the unit of energy on electricity bills, is equivalent to $3,6 \cdot 10^{6}\ \text{J}$.

1. a) Using $E = m \cdot c^2$ with $c = 3 \cdot 10^{8}\ \text{m/s}$, calculate what mass is equivalent to $1\ \text{kW} \cdot \text{h}$ of energy.
2. b) Compare that mass with that of a grain of sand ($10^{-5}\ \text{kg}$): how many times larger is the grain of sand? Comment on the result.

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 ($1\ \text{t} = 10^{3}\ \text{kg}$).

A pion has a proper lifetime of $2,6 \cdot 10^{-8}\ \text{s}$. In a linear accelerator, it travels at $v = 0,98c$ with $\gamma = 5$ (given: $c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) What is its mean lifetime measured in the laboratory ($\Delta t = \gamma \cdot \Delta t_0$)?
2. b) What average distance does it travel in the accelerator before decaying?
3. c) Explain why, without time dilation, pions would barely travel about $8\ \text{m}$, and what consequence that would have for their detection.

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 $100\ \text{W}$ light bulb remains turned on for $10$ years, that is, $3,2 \cdot 10^{8}\ \text{s}$.

1. a) Calculate the electrical energy consumed.
2. b) Using $E = m \cdot c^2$ with $c = 3 \cdot 10^{8}\ \text{m/s}$, calculate the mass equivalent to that energy.
3. c) Compare that mass with that of a bacterium ($10^{-15}\ \text{kg}$): how many times larger is it?

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 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.

In the Sun, each reaction of the proton–proton chain converts four protons into a helium nucleus with a mass defect of $\Delta m = 4,7 \cdot 10^{-29}\ \text{kg}$ ($c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the energy released in each reaction ($E = \Delta m \cdot c^2$).
2. b) The Sun radiates $3,9 \cdot 10^{26}\ \text{W}$: calculate how many of these reactions occur each second.
3. c) If each reaction consumes four protons of $1,67 \cdot 10^{-27}\ \text{kg}$ each, calculate the mass of hydrogen that the Sun fuses each second.
4. d) Compare the fused mass from part c) with the mass converted into energy from parts a) and b): what percentage of the mass is transformed into energy?

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.

One kilogram of petrol releases $4,6 \cdot 10^{7}\ \text{J}$ upon burning ($c = 3 \cdot 10^{8}\ \text{m/s}$).

1. a) Calculate the energy corresponding to the rest mass of $1\ \text{kg}$ according to $E = m \cdot c^2$.
2. b) How many kilograms of petrol would be equivalent to that energy?
3. c) What fraction of its rest mass does petrol convert into energy when burned?
4. d) In light of the result, explain why mass-energy equivalence goes unnoticed in chemical reactions and what type of processes do convert an appreciable fraction of mass into energy.

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.

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 $v = c \cdot \sqrt{1 - \frac{1}{\gamma^2}}$ 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 ($\Delta t = \gamma \cdot \Delta t_0$)?
4. d) In light of the results, evaluate the advantages and problems that interstellar travel at these speeds would present for human exploration.

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.

Unit challenge. Design a scientific report modelling the passage of cosmic muons through your own everyday environment.

1. Experimental measurement: Using a tape measure, measure the height of your room or of the stairs in your home ($L_0$). Perform 5 repetitions to calculate the mean value and its absolute uncertainty ($L_0 \pm \Delta L_0$).

2. Data processing and modelling: Assuming that a muon ($m = 1,88 \cdot 10^{-28}\text{ kg}$) travels that distance at different speeds ($v = 0,5c; 0,8c; 0,95c; 0,99c$), create a table and a plot showing length contraction ($L$), time dilation (travel time in your frame, $\Delta t = L_0/v$, and the muon's proper time, $\Delta \tau = \Delta t/\gamma$), and total relativistic energy ($E = \gamma m c^2$) versus $v/c$.

3. Critical analysis: Write a conclusion supporting Einstein's postulates, explaining why the particle cannot reach $c$ by means of the energy balance and contrasting the muon's perspective with your own.

Evaluation criteria:

• Precision in taking home-made measurements and correct expression of uncertainties.

• Mathematical rigour in calculating relativistic quantities and clarity in the plot.

• Coherent application of the postulates of relativity and the concept of relativistic energy.