Special Vs. Galilean Relativity Practice Problems

A futuristic magnetic levitation train weighing 4.5 × 10 ⁴ kg is moving at 0.81c (c is the speed of light). The rate of change of energy with respect to speed (dE/dv) represents the energy needed for each 1.0 m/s increase in speed. Determine the extra energy required to raise the train's speed by 1.0 m/s.

Specialized nano-probes developed for specific therapeutic applications have a rest half-life of 3.0 μs, but when accelerated to near-light speeds, their half-life extends to 6.0 μs. These nano-probes are 300 times as massive as a proton. What's the total energy of a fast-moving nano-probe?

In terms of c (the speed of light), at what speed does a nano-probe used in medical treatment have kinetic energy that is twice greater than what would be predicted by classical mechanics?

Consider a distant star named Zypher in a faraway galaxy, which emits energy through nuclear fusion at an astonishing rate of 4.8 × 10 ²⁶ W. Zypher is enormous, with a mass of 3.5 × 10 ³⁰ kg. Determine what percentage of Zypher's total mass is being converted into energy each year.

Consider a star whose mass is 1.78 × 10²⁹ kg (approximately equal to Sun). The star loses mass at a rate of 1.14 × 10¹⁹ kg per year due to fusion. It can sustain fusion until about 0.15% of its total mass is converted into energy and then stops. Determine the estimated lifetime of the star, expressed in million (10⁶) years.

Consider a hypothetical particle physics scenario where a newly discovered particle puyon moves to the right at 0.85c, and its antiparticle (anti-puyon) moves to the left at the same speed. Upon collision, they annihilate each other and produce four gamma-ray photons (equal energies). Determine the wavelength of these photons. The rest mass of a puyon (or anti-puyon) is approximately 1.53 × 10 ^-27 kg.

A man is on a train moving at 4.0 m/s. He can throw a ball at a speed of 7.0 m/s. What is the ball's speed relative to the ground if he throws the ball (i) in the direction the train is moving, (ii) in the opposite direction of the train's movement, and (iii) When the ball is thrown perpendicular to the direction of the train's movement.

A person stands on the rear end of a boat, moving away from the shore. A person can throw a frisbee at 25 m/s. They throw the frisbee towards the shore, and it approaches at a speed of 15 m/s. What is the boat's speed?