1. The escape speed V is the minimum speed needed to project a mass m upward fro

Question

1. The escape speed V is the minimum speed needed to project a mass m upward from the surface of another
mass M (such as a planet, asteroid, or star). It is given by using:

 

M = 1.99 × 1030 kg ,

G= gravitational constant 6.67 x 10-11 N*m2/kg2 

R = 6.96 × 108m.    is the radius of the surface from the center of M

 

v =2GM   ====>>>>>>   v= 6.1758 x 105 m/s         

          R              
a) Suppose the Sun mysteriously begins to contract. What would be the radius for which v escape fromthe surface equals the speed of light c? [This radius is known as the Schwarzschild radius. Anything thrown from that surface will eventually fall back on the surface. Interestingly Newton’s universal gravitation and Einstein’s general relativity predict the same value fo the Schwarzschild radius.]

b)  Find the Schwarzschild radius for Earth’s mass 5.97 × 1024 kg.

Explanation / Answer

(a) Mass of the sun, M = 1.99 x 10^30 kg Universal gravitational constant, R = 6.67 x 10^-11 Nm^2 / kg^2 Speed of light, C = 3 x 10^8 m/s Escape speed = Speed of light (C) [ for Schwarzschild radis ] V = [ 2 G M / R ] V^2 = 2 G M / R Schwarzschild radius, R = 2 G M / V^2
                                     = 2 * 6.67 x 10^-11 * 1.99 x 10^30 / ( 3 x 10^8 )^2                                       = 2.950 x 10^3 m                                       = 2.95 km (b) Mass of the earth, M = 5.97 x 10^24 kg Universal gravitational constant, R = 6.67 x 10^-11 Nm^2 / kg^2 Speed of light, C = 3 x 10^8 m/s Escape speed = Speed of light (C) [ for Schwarzschild radis ] V = [ 2 G M / R ] V^2 = 2 G M / R Schwarzschild radius, R = 2 G M / V^2
                                     = 2 * 6.67 x 10^-11 * 5.97 x 10^24 / ( 3 x 10^8 )^2                                       = 8.849 x 10^-3 m                                       = 8.85 mm Mass of the earth, M = 5.97 x 10^24 kg Universal gravitational constant, R = 6.67 x 10^-11 Nm^2 / kg^2 Speed of light, C = 3 x 10^8 m/s Escape speed = Speed of light (C) [ for Schwarzschild radis ] V = [ 2 G M / R ] V^2 = 2 G M / R Schwarzschild radius, R = 2 G M / V^2
                                     = 2 * 6.67 x 10^-11 * 5.97 x 10^24 / ( 3 x 10^8 )^2                                       = 8.849 x 10^-3 m                                       = 8.85 mm                                  

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