On April 13, 2029 (Friday the 13th!), the asteroid 99942 Apophis will pass withi

Question

On April 13, 2029 (Friday the 13th!), the asteroid 99942 Apophis will pass within 18600mi of the earth-about 1/13 the distance to the moon! It has a density of 2600kg/m^3 , can be modeled as a sphere 320m in diameter, and will be traveling at 12.6km/s.
(a)If, due to a small disturbance in its orbit, the asteroid were to hit the earth, how much kinetic energy would it deliver?
(b)The largest nuclear bomb ever tested by the United States was the "Castle/Bravo" bomb, having a yield of 15 megatons of TNT. (A megaton of TNT releases 4.184x10^15J of energy.) How many Castle/Bravo bombs would be equivalent to the energy of Apophis?

Explanation / Answer

mass of asteroid = density * volume = 2600 * (4/3) pi 160^3 = 4.461 x 10^10 kg

initial distance from center of earth = 6380 km + 18600 miles =

= 29927 km = 29927000 m

initial KE = (1/2) m v^2 = (1/2) * 4.461x10^10 * 12600^2 = 3.541 x 10^18 Joules

initial PE = -GMm / r= -6.67x10^-11*5.98x10^24*4.461x10^10 / 29927000 =

= -5.986 x 10^17

total initial energy = 3.541x10^18 - 5.986x10^17 = 2.946 x 10^18 Joules

final PE = -GMm / radius of earth = -6.67x10^-11*5.98x10^24*4.461x10^10 / 6380000 =

= -2.789 x 10^18 Joules

change in energy = final - initial = -2.789 - 2.946 = -5.735 x 10^18 Joules

KE delivered then would be 5.735 x 10^18 Joules

(b) convert to "bombs"

( 5.735x10^18 Joules ) * (1 bomb / 15 Mtons) * (1 Mton / 4.184x10^15 J) =

cancel units and

= 91.4 bombs

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