A Near-Earth Asteroid?. On April 13, 2029 (Friday the 13th!), the asteroid 99942 Apophis will pass within 18,600 mi of the earth—about the distance to the moon! It has a density of 2600 kg/m3, can be modeled as a sphere 320 m in diameter, and will be traveling at 12.6 km/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 4.184 × 1015 J of energy.) How many Castle/Bravo bombs would be equivalent to the energy of Apophis?

Solution 62P Here, we shall have to apply the formula for kinetic energy , which is K = mv …..(1), 2 2 where m is the mass of the object and v is its speed. In the question, we are given the density of the asteroid and its diameter. From these data, we can calculate its mass. Given, the diameter of Apophis is d = 320 m Its radius = 320/2 m = 160 m Therefore, the volume of Apophis is = × (160) m = 1.71 × 10 m 7 3 3 3 The density of Apophis = 2600 kg/m Therefore, its mass is m = 2600 kg/m × 1.71 × 10 m = 4446 × 10 kg = 4.44 × 10 10kg The speed of Apophis is v = 12.6 km/s = 12,600 m/s 1 2 (a) The kinetic energy possessed by Apophis is K = mv 2 10 2 K = 2 × 4.44 × 10 × (12,600) J K = 3.52 × 10 18 J 18 Therefore, Apophis would deliver an approximate kinetic energy of 3.520 × 10 J . 15 (b) A megaton of TNT releases 4.184 × 10 J of energy. Therefore, 3.52 × 10 18 J of energy is equivalent to 3.520×1018J 3 = 4.184×1015J = 0.84 × 10 = 840 megaton Now, given that one “ Castle/Bravo” bomb gives = 15 megaton Therefore, 840 megaton will be given by = 840/15 = 56 Castle/Bravo bombs. Therefore, 56 Castle/Bravo bombs will be equivalent to the energy of Apophis.