Suppose all the people of the Earth go to the North Pole and, on a signal, all jump straight up. Estimate the recoil speed of the Earth. The mass of the Earth is 6 1024 kg, and there are about 6 billion people (6 109). Take the average mass of a person to be 74 kg and the distance the average person's center of mass rises after leaving the ground to be 0.2 m.
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To estimate the recoil speed of the Earth when all the people jump at the North Pole, we can use the principle of conservation of momentum. The total momentum before the jump will be zero since the Earth is at rest. After the jump, the total momentum should also be zero to satisfy the conservation principle.
The formula to calculate momentum is:
Momentum = mass × velocity
Given that the mass of the Earth is 6 × 10^24 kg and there are about 6 billion people, each with an average mass of 74 kg, we can calculate the initial momentum of the people:
Initial momentum of people = Total mass of people × velocity of people
Total mass of people = Number of people × Average mass of a person
Now, we need to calculate the velocity of the people when they jump. We can find this by using the distance the average person's center of mass rises after leaving the ground (0.2 m) and the time it takes for this rise.
To find the time taken for the rise, we can use the formula:
time = distance / velocity
Assuming a constant acceleration, we can use the equations of motion:
distance = (1/2) × acceleration × time^2
velocity = acceleration × time
We know the distance (0.2 m), and acceleration due to gravity (approximately 9.8 m/s^2).
The time it takes for the rise can now be found by rearranging the equation for distance:
time = sqrt((2 × distance) / acceleration)
Substituting the values, we can calculate the time taken for the rise.
Now, we can plug in the calculated values into the equation for momentum to find the initial momentum of the people. Since momentum is conserved, the final momentum will also be zero. Therefore, the final momentum of the Earth must equal the initial momentum of the people.
Finally, we can rearrange the equation for momentum to solve for the recoil velocity of the Earth:
Recoil velocity of the Earth = (Initial momentum of the people) / (Mass of the Earth)
By substituting the calculated values into this equation, we can estimate the recoil speed of the Earth.