## Thursday, February 11, 2010

### Asteroid Attack!!!

Every few decades, an asteroid comes dangerously close to Earth’s orbit. While an impact with the Earth would surely cause widespread destruction of civilization and an end to life as we know it, this outcome may be preferable to sitting through Armageddon, Deep Impact, Meteor, or whatever is the next in a long line of horrendously clichéd movies about giant space rocks hurtling towards the Earth threatening to, once again, destroy life as we know it. But if an asteroid were about to hit the Earth, could we save our planet by all jumping simultaneously? If everyone on the planet gathered on the streets of New York and jumped, how fast would the Earth be pushed out of the way?

There are 6.7×109 people in the world and people weigh roughly 65 kg (~140 lbs) on average. This means the total mass of all humanity is given by,

total mass = (mass per person) · (# of people)
(65 kg per person) · (6.7×109 people)
= 4.4×1011 kg

If they all jump 0.3 m (~1.0 ft) off the ground, then we can compute the total potential energy from the jump using the formula below,

potential energy = (total mass) · (gravitational acceleration) · (height)
= (4.4×1011 kg) · (9.8 m/s2) · (0.3 m)
= 1.3×1012 kg.

Using the conservation of energy and the formula for kinetic energy, we can compute the speed at which they would leave the ground,

velocity = [2 · (kinetic energy) / (total mass)]1/2
= [2 · (1.3×1012 kg) / (4.4×1011 kg)]1/2
= 2.4 m/s

People leave the ground with a speed of about 2.4 m/s. The law of conservation of momentum states that when two things push off each other, the momentum of one thing is equal and opposite to that of the other thing. The momentum of an object is equal to its mass times its velocity, so we can compute the momentum of the people,

momentum = (mass) · (velocity)
= (1.3×1012 kg) · (2.4 m/s)
= 3.1×1012 kg·m/s.

The Earth’s mass is about 6.0×1024 kg. From this, we can calculate the speed at which the Earth will recoil,

Earth’s velocity = (momentum) / (mass)
= (3.1×1012 kg·m/s) / (6.0×1024 kg)
= 5.2×1013 m/s.

The Earth would only move at about 5.2×10-13 m/s. At that speed, the Earth would take an hour to move a distance of only a few atoms!

1. Could all 6 billion people even FIT on the streets of NYC? I'd start by approximating a person to be a cylinder 36 inches around.... surface area would be... 36 = 2*pi*r; r = 5.73; A = pi*r*r... about 103 square inches. Multiply by 6Billion == 6.18794419 × 10^11 square inches, or about 154 square miles.

Wikipedia says that the land area of NYC is ~304 sq miles, but most of that is buildings. My visual approximation is that buildings take up more than 50% of the space, so no, 6 billion of us couldn't fit on the streets, but as long as people were in lobbies of buildings & c, you *might* be able to squeeze all of us in. Maybe if you tried on New Years Eve...

Manhattan (also from Wikipedia) is only 23 sq miles of land, so I'd say you had absolutely no chance.

2. The energy you calculated is less than 1% of the largest recorded nuclear explosion (2.4*10^17 J).

Actually the Earth wouldn't move at all. Jumping off would push the Earth and people away from each other but then the gravitational force between the two would pull them right back to where they started.

More interesting question: If every person on Earth got onto the equator and started running in the same direction, how much would the Earth speed up or slow down?

4.4*10^11 kg of people. Usain Bolt ran the 100m in 9.54 s which gives an average speed of 10.4 m/s.

Angular momentum is equal to the moment of inertia multiplied by the angular velocity. Approximating the people as a continuous ring around the equator, the moment of inertia for a ring is
I = m*r^2 = 4.4*10^11 * 6378100^2 = 1.79*10^25 kg*m^2
The circumference of the Earth is about 4.01*10^7 m so running as fast as Usain Bolt gives an angular velocity of 1.64*10^-6 Hz. The total angular momentum of the sprinting world population is therefore 2.93*10^19 kg*m^2/s.

Similarly the moment of inertia of the Earth is
I = 2*m*r^2/5 = 2/5 * 5.97*10^24 * 6378100^2 = 9.72*10^37 kg*m^2
One sidereal day for the Earth is 86164 s giving an angular velocity of 7.29*10^-5 Hz. Therefore, the Earth's angular momentum is 7.09*10^33 kg*m^2/s.

At this point we could add or subtract the angular moment of the world population from that of the Earth depending on which way they were running (running with the Earth would slow it down while running against the Earth would speed it up) and back out the new angular velocity of the Earth. Unfortunately, it is very apparent that the Earth's spin rate would barely change because the Earth's angular momentum is many orders of magnitude larger than that of the Earth's population. Oh well.

Here's a question for you: If the human race used the planet Mercury as a giant flywheel and extracted energy out of the planet by despinning it, how many years could the planet satisfy all of our energy needs before it stopped spinning?