Escape velocity and black holes The work required to launchan object from the surface of Earth to outer space is given byw = JEF(x) dx, where R = 6370 km is the approximate radiusis the gravitational force between Earthand the object, G is the gravitational constant, M is the mass ofEarth, m is the mass of the object, and GM = 4 x 1014 m²/s?.GMmof Earth, F(x) =x?a. Find the work required to launch an object in terms of m.b. What escape velocity v, is required to give the object a kinetic1energy mv? equal to W?c. The French scientist Laplace anticipated the existence of blackholes in the 18th century with the following argument: If abody has an escape velocity that equals or exceeds the speedof light, c = 300,000 km/s, then light cannot escape the bodyand it cannot be seen. Show that such a body has a radiusRs 2GM/c. For Earth to be a black hole, what would itsradius need to be?

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Answered: Escape velocity and black holes The…

College Physics

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Chapter28: Special Relativity

Section: Chapter Questions

Problem 68PE: Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a)...

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Nuclear-powered rockets were researched for some years before safety concerns paramount. (a) What fraction of a rocket's mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of 1.00×105kg (100 tons), what total yield nuclear explosion in tons of TNT is needed?
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?
Our solar system orbits the center of the Milky Way galaxy. Assuming a circular orbit 30,000 ly in radius and an orbital speed of 250 km/s, how many years does it take for one revolution? Note than this is approximate, assuming constant speed and circular orbit, but it is representative of the time for our system and local stars to make one revolution around the galaxy.
Nuclear-powered rockets were researched for some years before safety concerns became paramount. (a) What fraction of a rocket's mass would have to be destroyed to get it into a low Earth orbit, neglecting the decrease in gravity? (Assume an orbital altitude of 250 km, and calculate both the kinetic energy (classical) and the gravitational potential energy needed.) (b) If the ship has a mass of 1.00105 kg (100 tons), what total yield nuclear explosion in tons of TNT is needed?
A muon formed high in Earth's atmosphere travels toward Earth at a speed v = 0.990c for a distance of 4.60 km as measured by an observer at rest with respect to Earth. It then decays into an electron, a neutrino, and an antineutrino. (a) How long does the muon survive according to an observer at rest on Earth? (b) Compute the gamma factor associated with the muon. (c) How much time passes according to an observer traveling with the muon? (d) What distance does the muon travel according to an observer traveling with the muon? (e) A third observer traveling toward the muon at c/2 measures the lifetime of the particle. According to this observer, is the muons lifetime shorter or longer than the lifetime measured by the observer at rest with respect to Earth? Explain.
The mass of a theoretical particle that may be associated with the uni?cation of the electroweak and strong forces is (a) How many proton masses is this? (b) How many electron masses is this? (This indicates how extremely relativistic the accelerator would have to be in order to make the particle, and how large the relativistic quantity (would have to be.)
A muon formed high in Earth's atmosphere travels toward Earth at a speed v = 0.990c for a distance of 4.60 km as measured by an observer at rest with respect to Earth. It then decays into an electron, a neutrino, and an antineutrino. (a) How long does the muon survive according to an observer at rest on Earth? (b) Compute the gamma factor associated with the muon. (c) How much time passes according to an observer traveling with the muon? (d) What distance does the muon travel according to an observer traveling with the muon? (e) A third observer traveling toward the muon at c/2 measures the lifetime of the particle. According to this observer, is the muons lifetime shorter or longer than the lifetime measured by the observer at rest with respect to Earth? Explain.
Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 19(N) nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?
(a) A panicle and its antiparticle are at rest relative to an observer and annihilate (completely destroying both masses), creating two y rays of equal energy. What is the characteristic y -ray energy you would look for if searching for evidence of proton-antiproton annihilation? (The fact that such radiation is rarely observed is evidence that there is very little antimatter in the universe.) (b) How does this compare with the 0.511-MeV energy associated with electron-positron annihilation?
An astronaut is traveling in a space vehicle moving at 0.500c relative to the Earth. The astronaut measures her pulse rate at 75.0 beats per minute. Signals generated by the astronauts pulse are radioed to the Earth when the vehicle is moving in a direction perpendicular to the line that connects the vehicle with an observer on the Earth. (a) What pulse rate does the Earth-based observer measure? (b) What If? What would be the pulse rate if the speed of the space vehicle were increased to 0.990c?
Collapse of a dust ball under Newtonian gravity. Suppose that a spherical ball of dust with no internal pressure and with an initial radius R0 and total mass M gravitationally collapses from rest. Show that the outer shell of the collapsing dust ball obeys the equation of motion Solve this equation of motion and show that the time to collapse is given by Hint: Write Newton’s law as . Let f(R) = dF(R)/dR and multiply both sides of by by . The resulting equation may be immediately integrated with respect to time by recognizing that . Integrating again with respect to time, this general technique yields t as a function of R where and C and C1 are constants determined by the initial conditions. Find the time of collapse of a dust ball with a mass and radius equal to that of the Sun. (RS = 6.96 × 1010 cm, MS = 1.99 × 1033 g, G = 6.67 × 10−8 cgs units) What prevents the Sun from collapsing in this way?
Unreasonable Results (a) Find the value of for the following situation. An Earth-bound observer measures 23.9 h to have passed while signals from a high-velocity space probe indicate that 24.0 h have passed on board. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

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