Question

The mass of a meteor with a radius of 1 km is about 9 x 1012 kg. The mass of a meteor also is proportional to the cube of its radius. Suppose a meteor with a radius of 8.8 km is moving at 1.7 x 104 m/s when it collides inelastically with the Earth. The Earth has a mass of 5.97 x 1024 kg and assume the Earth is stationary. The kinetic energy lost by the asteroid in this collision will be transferred to non-conservative work in heating the atmosphere and physically destroying the place where it lands. The Tsar Bomb, the largest atomic bomb ever tested, released 2.1 x 1017 J of energy. (Which, by the way, is 1000's of times more energy compared to the atomic bombs dropped in World War II.) How many MILLIONS of equivalent Tsar Bombs is the kinetic energy lost of this meteor?

Answer 1

m₁ = mass of meteor of radius "r₁" = 9 x 10¹² kg

r₁ = radius of meteor = 1 km

m₂ = mass of meteor of radius "r₂" = 9 x 10¹² kg

r₂ = radius of meteor = 8.8 km

since mass is propotional to cube of radius  

m₁/m₂ = (r₁/r₂)³

(9 x 10¹²)/m₂ = (1/8.8)³

m₂ = 6.13 x 10¹⁵ kg

v₂ = velocity of meteor before collision = 1.7 x 10⁴ m/s

M = mass of earth = 5.97 x 10²⁴ kg

v = velocity of earth before collision = 0 m/s

V = velocity of the combination after collision = ?

Using conservation of momentum

m₂ v₂ + M v = (M + m₂) V

(6.13 x 10¹⁵) (1.7 x 10⁴) + (5.97 x 10²⁴) (0) = ((6.13 x 10¹⁵) + (5.97 x 10²⁴)) V

V = 1.75 x 10^-5 m/s

kinetic energy lost is given as

E = (0.5) (m₂ v²₂ + M v² - (M + m₂) V²)

E = (0.5) ((6.13 x 10¹⁵) (1.7 x 10⁴)² + (5.97 x 10²⁴) (0)² - ((6.13 x 10¹⁵) + (5.97 x 10²⁴)) (1.75 x 10^-5)²)

E = 8.86 x 10²³ J

E = 8.86 x 10²³ J (1 Tsar bombs energy /(2.1 x 10¹⁷) J)

E = 4.22 x 10⁶ Tsar bombs energy

E = 4.22 millions of  Tsar bombs energy