Consequences of Relativity Practice Problems

Envision a hypothetical scenario where a novel particle, rutron, moves to the right and its antiparticle (anti-rutron) moves to the left with the same speed. When these particles inevitably collide, they annihilate each other, producing two gamma-ray photons of wavelength 1.2 fm and a viton and anti-viton pair with a speed of 0.92c. Determine the initial speed of the rutron and anti-rutron, in terms of c, before the collision. Take the mass of the rutron to be 1.52 × 10 ^-27 kg and that of the viton to be 8.21 × 10 ^-31 kg.

If a hypothetical particle, tepton, has a total energy of 3.0 GeV, calculate the speed (v) of the tepton (m = 9.11 x 10^-31 kg) in terms of the speed of light (c). Solve using the concepts of relativity.

Bob is located at a distance of x = 1242 m from the origin and when he observes light from two separate events. During Event "A", a flare gun fired at the origin (x = 0 m), and during Event "Q", another flare gun fired at x = 1850 m from the origin position. The light from both events reaches him simultaneously at 9.2 μs. Determine what the original time values are, considering when each of the flare guns were initially fired.

An aircraft flies 5900 km per day around Earth at 210 m/s. The aircraft returns to its starting point 4 days later at the same speed. How much has the pilot aged compared to their airbase colleagues? Hint: Use the binomial approximation.

For a positron (m = 9.109 × 10^-31 kg), at what kinetic energy does the Lorentz factor (γ) equal 1.44 in relativistic calculations? Express your final answer using the most appropriate prefixes: mega (M), giga (G), or tera (T).

The mass-energy equivalence principle allows us to convert mass into energy. Consider the following fission reaction: 

¹n₀ + ²³⁵U₉₂ → ²³⁶U₉₂ → ¹³⁷Te₅₂ + ⁹⁷Zr₄₀ + 2 ¹n₀

If, in this reaction, 0.211 u of mass is converted to kinetic energy, determine the total kinetic energy released.

According to quantum mechanics, electrons can have different orbital angular momenta, even if they have the same energy (state). What is the orbital quantum number l an electron could have in any chosen axis if the angular momentum vector L of the electron and the chosen axis has an angle of 24.4°? Assume the electron to be of a hydrogen atom and the angle to be smallest for the particular value of l.

Consider a particle physics experiment, where an electron gun accelerates electrons through a 2250 V potential difference towards some aluminum foil that has a mass of 8.0 mg. The electron beam causes a temperature increase of 9.0°C in the foil over a period of 13 seconds. Assuming that there is no energy loss by any other means and considering that the specific heat of aluminum is 903 J/kg•K, determine what the current of the electron beam will be.

In a high-energy physics experiment, a neutron is propelled to reach a velocity that is 0.998 times the speed of light. By what factor does the neutron's relativistic momentum surpass its classical (Newtonian) momentum?

In a physics experiment, a particle that has a mass of 2.30 grams is accelerated to reach a velocity that is 70% of the speed of light (0.70c). Determine (i) the rest energy, (ii) the kinetic energy, and (iii) the total energy of this particular particle at this speed. Express your final answer using the most appropriate of the following prefixes: mega (M), giga (G), or tera (T).

Consider a gourmet pizza that weighs 300 g and contains 600 food calories, which is equivalent to approximately 2.50 MJ of energy. Determine what the factor value will need to be for which the energy equivalent will exceed the dietary food energy provided by the pizza itself, as per Einstein's mass-energy equivalence principle.

In order to coordinate clocks in a reference frame, a light beam is activated at the origin when t = 0 s. What initial time value should be assigned to the timepiece positioned at the spatial coordinates (x, y, z) = (0 m, 35 m, 15 m)?

An alpha particle is observed to move in a straight line with a constant velocity of 4.5 × 10⁵ m/s when it passes through two parallel plates connected to a power supply, as shown below. Determine the magnitude and direction of the magnetic field localized within the region between the electrodes, enabling the charged particle to move without experiencing any deflection.

Consider two identical white dwarfs forming a binary system with a separation that equals four times their individual Schwarzschild radii. Determine their velocity with respect to their common center using Newtonian mechanics.

An alien civilization sends signals toward Earth, and these signals are detected as radio waves with frequencies about 10% lower than those measured on Earth. Estimate the speed of the alien civilization's planet using the Doppler effect for light. How would your answer differ if you used the classical Doppler effect?

An atomic species emits light with a characteristic frequency $f_{0}$ while at a rest state. A diatomic gas of such atomic species has been heated to temperature $T^{\prime}$. Due to their random thermal motions, some atoms move toward or against an onlooker. Assuming that root mean square speed characterizes their thermal motions:

(i) Determine the expression for the fractional variation $\frac{\Delta f}{f_{0}}$ between Doppler shifted frequencies corresponding to those moving straightly toward or against the onlooker, Assume $m c^{2} \gg 3 k T^{\prime}$.

(ii) Compute $\Delta f / f_{0}$ for a gas of oxygen atoms at 300 K.

Technological innovations in the alien's planet have facilitated the evolution of spacecrafts that move at nearly the speed of light. A 1.00 m long meter ruler, as measured by a scientist in a spacecraft is placed parallel to the direction of the spacecraft's motion. The spacecraft flies past the alien's planet at 82.5% of the speed of light. a) Find the ruler's length as measured by a scientist on the alien's planet. A scientist on the alien's planet measured the length of a different ruler in the spacecraft to be 1.00 m, b) what is the ruler's length measured by a scientist in the spacecraft?

A radioactive nuclide at a high altitude (h = 65.0 km) emits a particle that travels at 0.52689c toward the surface of a planet. Both h and speed are measured by a stationary observer on the planet's surface. i) Determine the time taken to cover h as measured by the observer on Earth. ii) Use relativity to determine the value of h as measured in the particle's inertial frame. iii) Use relativity to determine the time the particle takes, measured relative to its inertial frame. 

A shortlived particle created at an altitude of 38.0 km has an average lifespan of 2.20 microseconds. The particle moves at a speed of 0.996c. i) Compute the maximum distance covered by the particle during its lifespan. ii) According to the frame of the particle, its lifespan is 2.20 microseconds. Determine the altitude it has to descend measured relative to its frame. iii) Interpret what the result in part i) and ii) imply regarding the possibility of the particle covering the entire altitude to the Earth's surface. 

An electromagnetic wave's emitter moves in a straight line relative to a detector. The detector records a frequency that is 3/4 times the frequency measured in the emitter's rest frame. i) Determine the speed of the emitter relative to the detector. ii) What is the direction of motion (toward or away from) relative to the detector? 

An airplane is moving toward a green signal light. Green light has a wavelength of 550 nm. A crew member in the airplane views the light to be blue (wavelength  = 495 nm). What is the speed of the airplane in terms of the speed of light? The patent owner is to be rewarded for inventing the high-speed plane. Determine the reward (in $) earned by the patent owner.  The reward is calculated as $2.50 for every kilometer per hour above the usual cruising speed of 920 km/h.

In a remote galaxy, astronomers detect a blazar receding from Earth at a velocity of 0.75c. Along the same line of sight, a galaxy closer to Earth is also moving away at a speed of 0.30c. Determine the blazar's recessional speed, expressed as a fraction of the speed of light (c), as measured by astronomers in the nearby galaxy.