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!