Asteroid split lab

Question

Background: The two tables below contain both the assumptions and parameters needed to evaluate if the plucky heroes in Armageddon could save Earth. In order to miss, Earth two conditions must exist.

1) The nuclear bomb must actually split the asteroid. We will assume that this happens and uses up none of the bomb blast’s energy.

2) The separation velocity of the asteroid halves has to be high enough to separate each of them by the radius of Earth plus 1290 kilometers (according to the movie) in the time it takes to reach Earth after the nuclear blast.

table 1) assumptions to be used for analysis

-Diameter of asteroid: 1000 km

-Density of asteroid: 7000 kg/m^3

-Shape of asteroid: Sphere

-Nuclear bomb's yield: 100 megatons

-Amount of Thermonuclear Energy Converted to the kinetic energy of the asteroid pieces:100%

table 2) Parameters to be used for the analysis

Velocity of Asteroid (toward Earth): 10,000 m/s

Distance from Earth that the bomb is detonation: 101,000 km

Earth’s Radius: 6.38 x 10^6 m

Factor for converting megatons of TNT to joules: 1 megaton of TNT = 4.184 x 10^15 Joules

Questions:

1- Assuming that 100% of the nuclear bomb’s explosive energy is converted to kinetic energy and that when the bomb explodes the asteroid is split into equal halves, determine if it will strike earth.

First draw an annotated diagram of the sequence of events.

2- Conclusions: Should we hire miners to save the planet if a larger asteroid is detected to be coming towards Earth?

3- Bonus: based on the information on the previous page, could superman have saved earth? Superman can generate a force of 3.92x10^24

Answer 1

Let

r = 500 km = 5×10⁵m be the radius of the asteroid,

d = 7×10³ kg/m³ be the density of the asteroid,

E = 100×4.184×10¹⁵ J = 4.184×10¹⁷ J be the energy released by the bomb,

v = 10⁴ m/s be the velocity of the asteroid towards the Earth,

s = 101,000 km = 101×10⁶ m be the distance from detonation to Earth,

R = 6.38×10⁶ m be the radius of the Earth.

In the spirit of the given simplifying assumptions I will make more simplifying assumptions:

- Earth's gravity does not effect the asteroid,

- All the bomb's energy is converted into kinetic energy of the two halves,

pushing them in a direction exactly orthogonal to the asteroid's trajectory.

The time it take the asteroid (or its pieces) to reach the Earth is

T = s/v = 101×10⁶/10⁴ = 10100 s

The volume of the asteroid is

V = 4πr³/3

The mass of the asteroid is

m = Vd = 4πr³d/3 =

4×π×(5×10⁵)³×7×10³/3 = 3.665×10²¹ kg

The mass of each of the asteroid's halves is m/2.

Let w be the orthogonal velocity that the bomb imparts on each half.

The kinetic energy of each half (in the perpendicular direction) is (m/2)w²/2

This kinetic energy is equal half the bomb's yield:

mw²/4 = E/2

w = √(2E/m) = √(2×4.184×10¹⁷/3.665×10²¹) = 0.015 m/s

In time T each half covers the distance

Tw = 10100×0.015 = 152.6 m

Since 152.6 m is much smaller that the radius of the Earth,

the asteroid will strike the Earth.

Question 2:

Yes, we should hire miners, because in any hopeless situation the most valuable commodity is hope,

no matter how false.

But the give people hope we must stop teaching physics ;-)

Question 3:

Let

F = 3.92×10²⁴ be the Superman's force.

I will assume that he pushes the asteroid in a direction perpendicular to its trajectory.

It gives the asteroid perpendicular acceleration

a = F/m

In time T, the asteroid will cover perpendicular distance

aT²/2 = FT²/2m =

3.92×10²⁴×10100²/2×3.665×10²¹ = 5.5×10¹⁰ m

That gives must greater clearance than the radius of Earth -- 6,380 km -- plus the extra distance of 1290 km.