Exam 26: The Special Theory of Relativity

An unstable particle is moving at a speed of $1.6 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ relative to a laboratory. Its lifetime is measured by a stationary observer in the laboratory to be $6.8 \times 10 ^ { - 6 } \mathrm {~s}$ . What is the lifetime of the particle as measured in the rest frame of the particle? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

Two stars are at rest relative to each other. A spaceship is moving between these two stars at 0.932c relative to each star. To someone in the ship, the distance between the two stars appears to be 26)4 light-years. What is the distance between the stars in their own rest frame?

What is the total energy of an electron moving with a speed of $0.950 c ? \left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } , m _ { e l } = 9.11 \right.$ $\times 10 ^ { - 31 } \mathrm {~kg}$ )

A starship is constructed to be 100 m long in the factory. It is launched and as it coasts by Earth, it is measured to be shortened to 5.48 m long. (a) How fast, in terms of the speed of light, is it moving past Earth? (b) What length does the ship's crew measure for the ship?

A $1.0 - \mathrm { kg }$ aluminum ingot is heated from $0 ^ { \circ } \mathrm { C }$ to $100 ^ { \circ } \mathrm { C }$ . By what amount does its mass change? The specific heat of aluminum is $900 \mathrm {~J} / \mathrm { kg } \cdot \mathrm { K } . \left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

A 30-year-old astronaut goes off on a long-term mission in a spacecraft that travels at speeds close to that of light. The mission lasts exactly 20 years as measured on Earth. Biologically speaking, at the end of the mission, the astronaut's age would be

An electron has a speed of 0.783c. Through what potential difference would the electron need to be accelerated from rest in order to reach this speed? The rest mass energy of an electron is 0.511 MeV)

A spaceship is moving away from an asteroid with a speed of 0.80c relative to the asteroid. The spaceship then fires a missile with a speed of 0.50c relative to the spaceship. What is the speed of The missile measured by astronauts on the asteroid if the missile is fired away from the asteroid?

How many joules of energy are required to accelerate a $1.0 - \mathrm { kg }$ rock from rest to a speed of $0.866 c$ ? ( $c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ )

If the kinetic energy of a particle is $3.00$ times its rest energy, the speed of the particle is closest to which of the following values?

How much work must be done to accelerate a particle of mass $3.5 \times 10 ^ { - 14 } \mathrm {~kg}$ from a speed of $1.2 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ to a speed of $1.9 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ ? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

The figure shows a right-angled construction frame $\mathrm { ABC }$ . When measured at rest, it has dimensions as follows: $\mathrm { AB } = 26.0 \mathrm {~m} , \mathrm { AC } = 10.0 \mathrm {~m}$ , and $\mathrm { BC } = 24.0 \mathrm {~m}$ . This frame is now given a new velocity of $0.760$ c, relative to Earth, in a direction parallel to AC. How long is part AC of this frame with the new velocity as measured by Earth-based engineers?

Two spaceships, each 100 m long in its own rest frame, approach Earth from opposite directions, each with a speed 0.50c relative to Earth. As measured by a passenger in one of the spaceships, How long is the other spaceship?

A certain unstable particle has a mean lifetime of $1.52 \times 10 ^ { - 6 } \mathrm {~s}$ . This is the time interval between its creation in a nuclear process and its extinction into decay products, as measured in a frame of reference at rest with respect to the particle. An average such particle is observed by a scientist on Earth to travel $342 \mathrm {~m}$ in its lifetime. What is the speed of the particle relative to Earth? $( c = 3.00 \mathrm { x }$ $10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ )

When a proton is traveling at 0.80c, what is the ratio of its kinetic energy to its rest energy?

The total energy of a moving electron is $0.758 \mathrm { MeV }$ . What is the speed of the electron in terms of $c$ ? The rest mass energy of an electron is $0.511 \mathrm { MeV }$ .

A spaceship traveling at constant velocity passes by Earth and later passes by Mars. In which frame of reference is the amount of time separating these two events the proper time?

One 20-year-old twin brother takes a space trip with a speed of 0.80c for 30 years according to a clock on his spaceship. At the end of this trip, what are (a) his own age and (b) the age of his Earth-based twin brother?

A spacecraft is measured by a scientist on the ground to have a length of $81 \mathrm {~m}$ as it flies overhead with a speed $2.4 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s }$ . The spacecraft then lands and its length is again measured by the scientist at rest on the ground. What result does he now get for the Iength of the spacecraft? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$

A proton, which has a mass of $1.67 \times 10 ^ { - 27 }$ , is moving at $0.80 c$ relative to the laboratory. What does a physicist in that lab measure for its momentum? $\left( c = 3.0 \times 10 ^ { 8 } \mathrm {~m} / \mathrm { s } \right)$