This problem deals with escaping the Earth and failing to escape from a black ho

Question

This problem deals with escaping the Earth and failing to escape from a black hole.

(a) A cannon is located on the surface of the Earth. It is pointed straight up (directly away from the center of the Earth), and a cannonball with mass 10 kg is fired. What is the minimum initial velocity of the cannonball such that it can escape the gravitational pull of the Earth and never return? This velocity is called the escape velocity. Please answer symbolically before substituting numbers.

(b) A body of mass M will act as a black hole if its radius is less than or equal to a critical radius RS, from within which not even light can escape. The radius RS is called the Schwarzchild radius, and a simple calculation using Newtonian mechanics gives the correct answer by setting the escape velocity equal to the speed of light c. Find an expression for RS. (It turns out, the correctness of this result is partly accidental, since a proper calculation requires using Einstein’s theory of general relativity.) For a burned-out, collapsed star whose mass is 5 times the mass of our sun, what is the value of RS (write your answer in kilometers)?

Explanation / Answer

a)

The Law of Conservation of Energy states that the total energy of a closed system remains constant. In this case, the closed system consists of the two objects with the gravitational force between them and no outside energy or force affecting either object.

Thus the total final energy- potential energy plus kinetic energy—must equal the total initial energy:

TEi = TEinfinity

KEi + PEi = 0

Substitute values:

mve2/2 GMm/Ri = 0

Add GMm/Ri to both sides of equation:

mve2/2 = GMm/Ri

Solve for ve2

ve2 = 2GM/Ri

Take the square root of each expression to get,

ve = ± sqrt(2GM/Ri)

Considering our gravitational convention for direction, ve is upward or away from the other object and is thus negative,

ve = - sqrt(2GM/Ri) -------------(1)

b)

From (1),

ve = sqrt(2GM/R)

c= sqrt(2GM/Rs)

c2 = 2GM/Rs

Rs = 2GM/c2

Now M= 3*msun = 5*1.989*10^30 kg

Rs = 2GM/c2 = (2*6.67*10^-11*5*1.989*10^30)/(3*10^8)^2 = 14740.7 m

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