91Ó°ÊÓ

The Conservation of Momentum is a fundamental principle in physics that states the total momentum of an isolated system remains constant if no external forces act upon it. Momentum, a product of an object's mass and velocity, is an essential concept that helps us understand motion and interactions. In the given exercise, when 300 million people drop from a height of 1 meter simultaneously, they impart a certain momentum to Earth. Since before the fall, the momentum of the system (Earth plus people) was zero, the momentum after the fall must also be zero. This means the Earth gains momentum equal in magnitude but opposite in direction to the total momentum of the people.

• No external forces: The absence of any net external force allows the conservation principle to hold.
• Closed system: Earth and the people are considered a single system during their interaction.
• Equal and opposite momentum: The changes by the individuals are counterbalanced by an opposite change in the Earth, maintaining conservation.

Kinematics is the branch of physics that describes the motion of objects without considering the forces causing the motion. It helps us determine how fast something is moving based on initial conditions and physical laws.In this scenario, kinematics helps calculate the velocity of each person as they fall from a height of 1 meter. The velocity of a free-falling object can be determined using the kinematics equation for velocity:\[ v = \sqrt{2gd} \]where:

• \( v \) is the final velocity,
• \( g \) is the acceleration due to gravity, approximately \( 9.81 \mathrm{~m/s^2} \), and
• \( d \) is the distance fallen (1 meter in this case).

Using this formula, each person reaches a velocity of approximately \( 4.43 \mathrm{~m/s} \) as they touch the ground. Understanding this velocity is crucial for solving the problem since it contributes directly to the overall momentum calculation.

Calculating velocity is key in many physical problems to understand how objects will behave. In this exercise, the velocity calculation helps determine the speed at which the Earth would theoretically move as a consequence of the people's fall.The momentum imparted by each person is determined using:\[ p = mv \]where:

• \( m \) is the mass of the person, 65 kg in this scenario, and
• \( v \) is the velocity computed through kinematics (i.e., \( 4.43 \mathrm{~m/s} \)).

The result of this calculation is a momentum per person of \( 287.95 \mathrm{~kg~m/s} \). Multiplying this by 300 million people gives a total momentum of \( 8.64 \times 10^{10} \mathrm{~kg~m/s} \) which is then used to estimate Earth's change in speed. By dividing this value by Earth's mass \( 5.972 \times 10^{24} \mathrm{~kg} \), it turns out that the Earth's speed changes by a tiny \( 1.446 \times 10^{-14} \mathrm{~m/s} \), too small to notice in practice.