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The common question "What would happen if everyone in the world got together and jumped at once?" has been asked by many physicists and is very popular. To answer this question, we would need to assume that the entire Earth's population was transported to one place. Other assumptions include that there would 7 billion people in that one place with an average weight of 50 kg and an average vertical jump of .3 meters. Also, the mass of earth is 6 x 1024 kg and force of gravity near the Earth’s surface is 9.8 N. With these assumptions there are not outside forces on the system meaning that both energy and momentum would be conserved in the equation. This means that the initial momentum of the planet plus the initial momentum of the people must equal the final momentum of the planet plus the final momentum of the people. Let’s say that the initial instance is right after the people jump and that the final instance is when the people are at their maximum height. Energy is also conserved and therefore there is both kinetic energy and gravitational potential energy in this equation written below.
It is important that at the initial position there is no gravitational potential energy even though the people are moving and at the final position the people and the earth are .3 meters apart and not moving. This makes the problem a one-dimensional problem where the y-direction is the direction the people jump. From here we can modify the conservation of momentum equation:


Then we must modify the conservation of energy equation to get the initial velocity


Next, we must combine the two equations of momentum and energy together and get the velocity of the earth on one side.


Finally, we must solve for the velocity of the Earth and plug in the values from above to get speed of the earth after the people jump. This makes sense as the mass of the Earth is so massive that it would be very difficult for anything to create a noticeable change in speed.

 
The people of earth now have a much more pressing issue on their hands then the slight change in the speed of the earth that their efforts just created. They have all been transported into an area roughly the size of Rhode Island for them to packed tightly enough for this math to make sense. For convenience sake let’s just say they were transported to the state of Rhode Island. After the sound waves from that many feet hitting the ground at once dissipate, the people look around in confusion asking why they had just came here to jump once. The people’s efforts to contact their friends and loved ones using their cell phones end in failure when the tremendous load causes the local cell networks to collapse. The rest of the world begins to grind to a halt without humans. T.F. Green, the main airport in Rhode Island attempts to get people home by consolidating fuel and planes to return the people, but even if the airport managed to achieve 500% capacity it would have to run for centuries to return all the people. People try to escape Rhode Island in all sorts of means, by container ship, car, and rail but since so much of society has shut down without human intervention there are many complications in getting the people out of Rhode Island. The meager food and water in Rhode Island intended for its small population of one million people is quickly consumed and the state turns into a warlord infested violence ridden locale. In this desperate situation, billions of people die from lack of food and water. The survivors struggle back into their life and humanity survives having answered the question of what happens when everyone jumps at once but at the terrible cost in human lives.   




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