A Near Miss! In the early morning hours of June 14, 2002, the Earth had a remark

Question

A Near Miss! In the early morning hours of June 14, 2002, the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated 2002 MN, remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was 73600 miles from the center of the Earth-about a third of the distance to the Moon.

Part A:Find the speed of the asteroid at closest approach, assuming its speed at infinite distance to be zero and considering only its interaction with the Earth.

Part B:Observations indicate the asteroid to have a diameter of about 2.0

Explanation / Answer

a)

use conservation of energy; at infinity, it had zero total energy, so must have zero total energy throughout its journey

at a distance of r from the earth, the potential energy due to the earth's gravitational field is

PE = - GMm/r

G=newtonian grav constant, M, m are the masses of the earth and the asteroid, r is the distance from the center of the earth

the loss in PE must equal the gain in KE, or

1/2 m v^2 = GMm/r

so that v=Sqrt[2GM/r]

G=6.67x10^-11 in MKS
M=5.98x10^24kg
r=73600 miles = 118,500km = 1.18x10^8m

therefore, v = 2590m/s

b)

Radius r = 2km/2 = 1000m

Volume = (4/3)*pi*r^3 = 4.189*10^9 m^3

Density = 3g/cm^3 = 0.003kg/cm^3 * 10^6cm^3/m^3 = 3000 kg/m^3

Mass m = Volume * Density = 1.257*10^13 kg

KE = 0.5*m*v^2 = 0.5*1.257*10^13 * 2590^2 = 4.2*10^19 J

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