Calculus and Above

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1. The work required to...

1. The work required to launch an object from the surface of Earth to outer space is given y W=integral from R to infinity F(x)dx, where R= 6370 km is the approximate radius of Earth, F(x)= GMm/x^2 is the gravitational force between Earth and the object, G is the gravitational constant, M is the mass of the Earth, m is the mass of the object, and GM=4*10^14 m^3/s^2.

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 kinetic energy 0.5*m*v^2 equal to W?

c. The French scientist anticipated the existence of black holes in 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 the light cannot escape the body and it cannot be seen. Show that such a body has a radius R<=2 GM/c^2. For Earth to be a black hole, what would its radius need to be?

"I need only part c."

2. find integral

integral from 1 to infinity dx/(e^(x+1)+e^(3-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 kinetic energy 0.5*m*v^2 equal to W?

c. The French scientist anticipated the existence of black holes in 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 the light cannot escape the body and it cannot be seen. Show that such a body has a radius R<=2 GM/c^2. For Earth to be a black hole, what would its radius need to be?

"I need only part c."

2. find integral

integral from 1 to infinity dx/(e^(x+1)+e^(3-x))

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