Iodine (¹³¹I), which is initially at rest, decays into xenon (¹³¹Xe) through the emission of a beta particle, which in this case is an electron. The decay is as follows:

¹³¹I → ¹³¹Xe + e⁻

Formulate an equation for the conservation of relativistic energy in this decay in terms of masses of iodine (mᵢ), xenon (mₓ), and the electron (mₑ), along with their relativistic factors: xenon (γₓ) and the electron (γₑ).

Assume a board is a box of width 2.1 m and a sphere of mass 165 g is treated like a particle inside the box. Assume the sphere moves by sliding with negligible friction such that it has zero rotational kinetic energy. i) determine the least energy of the particle. ii) Using the equation for translational kinetic energy, determine the speed of the particle iii) determine the time required for the sphere to cover the width of the board.

Astrophysicists on Earth detect two significant astronomical events occurring 15 years apart at two different positions in space, separated by 75 light-years. A telescope, moving in space, measures the distance between the two events to be 95 light-years. Determine the time between the two events as it would be measured by the telescope.

A galaxy has a radius of 30000 light-years (ly). A spacecraft traveling through the galaxy perceives the radius of the galaxy to be 125 ly. Determine the time needed, as measured in the galaxy's frame, for the spacecraft to cross the entire galaxy.

Imagine that a spaceship is traveling through space, and that the captain measures the length of their spaceship to be 135 m. However, when the spaceship is in close proximity to the planet Mars, a scientist on Mars measures the length of the spaceship to be 114 m. Calculate the speed of the spaceship in terms of c.

A rectangular prism has a length of 50 cm, a width of 25 cm, and a height of 10 cm when it is at rest with respect to an inertial frame, S. The density of the rectangular prism is 2640 kg/m ³. Determine the density of the rectangular prism when it is moving in a straight line parallel to its length at a speed of 0.75c with respect to the frame S.

Consider a positron moving at a constant speed of 0.99999965 c along a 4.5 km linear underground tunnel. What is the length of the tunnel as measured in the reference frame of the positron?