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Escape velocity and black holes The work required to
Chapter 7, Problem 73E(choose chapter or problem)
Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by \(W=\int_{R}^{\infty} F(x) \ d x\) , where R =6370 km is the approximate radius of Earth, \(F(x)=G M m / x^{2}\) is the gravitational force between Earth and the object, G is the gravitational constant. M is the mass of Earth, m is the mass of the object, and \(G M=4 \times 10^{14} \mathrm{\ m}^{3} / \mathrm{s}^{2}\).
a. Find the work required to launch an object in terms of m.
b. What escape velocity \(v_{e}\) is required to give the object a kinetic energy \(\frac{1}{2} m v_{e}^{2}\) equal to W?
c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light. c =300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius \(R \leq 2 G M / c^{2}\). For Earth to be a black hole, what would its radius need to be?
Questions & Answers
QUESTION:
Escape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by \(W=\int_{R}^{\infty} F(x) \ d x\) , where R =6370 km is the approximate radius of Earth, \(F(x)=G M m / x^{2}\) is the gravitational force between Earth and the object, G is the gravitational constant. M is the mass of Earth, m is the mass of the object, and \(G M=4 \times 10^{14} \mathrm{\ m}^{3} / \mathrm{s}^{2}\).
a. Find the work required to launch an object in terms of m.
b. What escape velocity \(v_{e}\) is required to give the object a kinetic energy \(\frac{1}{2} m v_{e}^{2}\) equal to W?
c. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light. c =300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius \(R \leq 2 G M / c^{2}\). For Earth to be a black hole, what would its radius need to be?
ANSWER:Problem 73EEscape velocity and black holes The work required to launch an object from the surface of Earth to outer space is given by , where R =6370 km is the approximate radius of Earth, is the gravitational force between Earth and the object, G is the gravitational constant. M is the mass of Earth, m is the mass of the object, and GM = 4 × 1014 m3/s2.a. Find the work required to launch an object in terms of m.b. What escape velocity is required to give the object a kinetic energyequal to Wc. The French scientist Laplace anticipated the existence of black holes in the 18th century with the following argument: If a body has an escape velocity that equals or exceeds the speed of light. c =300,000 km/s, then light cannot escape the body and it cannot be seen. Show that such a body has a radius R 2GM/c2. For Earth to be a black hole, what would its radius need to beSolution:Step 1To find the work required to launch an object in terms of m.The work required to launch an object from the surface of Earth to outer space is given by , where . Putting in the value of GM and R, we get .So, work done to launch an object of mass m is .