An Extra Jump

By: Catherine Medeiros

The inner child in all of us has all though about it. What if every person in the world (yes, all 7 billion of them), gathered in one singular location and jumped all at the exact same instant. What would happen? Well, let's say these 7 billion people that weigh a total of approximately 363 billion kilograms, all gathered in Rhode Island. Everybody would fit comfortably and take up about the same area as RI. Everybody would be close enough where the jumping may have an affect on the Earth and nobody would be crammed together.

The clock strikes, say, 3:00 pm and everbody jumps. The average human vertical jump is a little under a half a meter, 0.4 meters. We can start with our basic energy and momentum equations.

Pi = Pf -> Pearth + Ppeople = Pearthf + Ppeoplef

Ei = Ef -> 1/2mv^2(earth) + 1/2mv^2(people) = 1/2mv^2(earthf) +1/2mv^2(peoplef)

The next step is to find the final velocity of the Earth. One can do this by using the momentum equation in the y direction.

mv(earth) + mv(people) = mv(earthf) + mv(peoplef)

Neither the Earth nor the people are moving in the y direction before the jump, so the right side of the equation will equal zero. 

0 = mv(earthf) + mv(peoplef)

-mv(peoplef) = mv(earthf)

-mv(peoplef) = v(earthf)

m(Earth)lkdjflsjfsks

Okay so, there is absolutely no pretty way to do this. If I were to use substitution, I would have what I am trying to solve for in the equation. I think that the best way to solve for the final velocity of the Earth is to first solve for the final velocity of the people by plugging in the final velocity of the Earth that I just found. 

 1/2mv^2(earth) + 1/2mv^2(people) = 1/2mv^2(earthf) +1/2mv^2(peoplef)

 1/2mv^2(earth) + 1/2mv^2(people) = 1/2m(-mv(peoplef)/m(Earth))^2 +1/2mv^2(peoplef)

 1/2mv^2(earth) + 1/2mv^2(people) = (-m^2v(peoplef))^2 + 1/2mv^2(peoplef)

aldkfjadklfjasdjfaslm(Earth)

m(Earth)(1/2mv^2(earth) + 1/2mv^2(people)) = (-m^2v(peoplef))^2 + 1/2mv^2(peoplef)

m(Earth)(1/2mv^2(earth) + 1/2mv^2(people)) = m^4v(peoplef)^2 + 1/2mv^2(peoplef)

m(Earth)(1/2mv^2(earth) + 1/2mv^2(people)) = v(peoplef)^2(m(earth)^4 + 1/2m(people))

m(Earth)(1/2mv^2(earth) + 1/2mv^2(people)) = v(peoplef)^2

(m(earth)^4 + 1/2m(people))adjlakjflaakldfja

SQUARE ROOT m(Earth)(1/2mv^2(earth) + 1/2mv^2(people)) = v(peoplef)

(m(earth)^4 + 1/2m(people))

Yikes, that is what teh final velocity of all the people in the world jumping would equal. We can now take this and plug it into the equation we had above. 

-mv(peoplef) = v(earthf)

m(Earth)lkdjflsjfsks

-m(SQUARE ROOT m(Earth)(1/2mv^2(earth) + 1/2mv^2(people)) / (m(earth)^4 + 1/2m(people)))

m(Earth)lkdjflsjfsks

Okay, that is a lot messier than I thought. BUT, we have all these values. 

Mass of people = 52kg(7billion) = 3.57 x 10^11 kg

Mass of Earth = 5.972 x 10^24 kg

Velocity of the Earth (normal x direction) = 460 meters/second

Velocity of the people (normal x direction) = 0

Now, we just have to plug these values into our calculator. I know its hard to tell here, but the top numerator of the fraction is a fraction as well. Therefore, we have a complex fraction. No fear, calculators were made for a reason. 

Drum roll, the answer is 0. 

Now, either Desmos.com is broken, I did my math wrong (a very real possibilty) or I did it right and the Earth really won't move. 

While we all jump, loads of energy is transferred to the Earth when we land. However, the surface of the Earth absorbs this energy and the jumping really doesn't have any affect on us. 

Except the fact that there is 7 billion people standing in the smallest state in The United States of America. 7. Billion. People. Not everybody can drive back home, many have to take a plan. TF Green would be overwhelmed with passengers. TSA would become so busy, that they would not be able to do a complete check of every person. With the entire world in one state, terrorist groups are inevitably going to be there as well. With TSA overwhelmed, things could go south extremely quickly. Complete chaos. All highways that leave the state would soon be a like a parking lot full of cars which would no doubt be the biggest traffic jam in the history of the world. Madness would break out in our tiny state. However, when all is said and done, people would have to stay and buy things in RI, helping our economy greatly.




Sources:
https://www.livescience.com/33383-everyone-on-earth-jumping-at-once.html
https://en.wikipedia.org/wiki/Earth_mass
https://www.scientificamerican.com/article/how-fast-is-the-earth-mov/