5.9 Relativistic energy  (Page 11/16)

Repeat the preceding problem with the ship heading directly away from Earth.

If a spaceship is approaching the Earth at 0.100 c and a message capsule is sent toward it at 0.100 c relative to Earth, what is the speed of the capsule relative to the ship?

(a) Suppose the speed of light were only 3000 m/s. A jet fighter moving toward a target on the ground at 800 m/s shoots bullets, each having a muzzle velocity of 1000 m/s. What are the bullets’ velocity relative to the target? (b) If the speed of light was this small, would you observe relativistic effects in everyday life? Discuss.

If a galaxy moving away from the Earth has a speed of 1000 km/s and emits 656 nm light characteristic of hydrogen (the most common element in the universe). (a) What wavelength would we observe on Earth? (b) What type of electromagnetic radiation is this? (c) Why is the speed of Earth in its orbit negligible here?

A space probe speeding towards the nearest star moves at $0.250c$ and sends radio information at a broadcast frequency of 1.00 GHz. What frequency is received on Earth?

Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?

(a) Calculate the speed of a $1.00\text{-}\text{μg}$ particle of dust that has the same momentum as a proton moving at 0.999 c . (b) What does the small speed tell us about the mass of a proton compared to even a tiny amount of macroscopic matter?

(a) Calculate $\gamma$ for a proton that has a momentum of $1.00\text{kg}·\text{m/s}\text{.}$ (b) What is its speed? Such protons form a rare component of cosmic radiation with uncertain origins.

Show that the relativistic form of Newton’s second law is (a) $F=m\frac{du}{dt}\frac{1}{{\left(1-u^2\text{/}{c}^2\right)}^{3\text{/}2}};$ (b) Find the force needed to accelerate a mass of 1 kg by 1 ${\text{m/s}}^2$ when it is traveling at a velocity of c /2.

A positron is an antimatter version of the electron, having exactly the same mass. When a positron and an electron meet, they annihilate, converting all of their mass into energy. (a) Find the energy released, assuming negligible kinetic energy before the annihilation. (b) If this energy is given to a proton in the form of kinetic energy, what is its velocity? (c) If this energy is given to another electron in the form of kinetic energy, what is its velocity?

What is the kinetic energy in MeV of a π-meson that lives $1.40\times 10^{\mathrm{-16}}\text{s}$ as measured in the laboratory, and $0.840\times 10^{\mathrm{-16}}\text{s}$ when at rest relative to an observer, given that its rest energy is 135 MeV?