A 90 kg person stands on the ground before jumping vertically upward. If momentum is truly conserved in all closed isolated systems, then the Earth-person system should conserve momentum. If the person jumps with a speed of 3.5 m/s, what is resulting speed of the Earth in the opposite direction? (Earth's is approximately 5.96 • 1024 kg)
To solve this problem, we can use the concept of conservation of momentum. According to the law of conservation of momentum, the total momentum before an event should be equal to the total momentum after the event, as long as no external forces act on the system.
Let's first find the initial momentum of the system, which consists of the Earth and the person. The momentum of an object is calculated by multiplying its mass by its velocity.
The mass of the person is given as 90 kg, and the velocity before jumping is 0 m/s. Therefore, the initial momentum of the person is:
Momentum_person = mass_person × velocity_person
= 90 kg × 0 m/s
= 0 kg·m/s
Now, let's calculate the initial momentum of the Earth. The mass of the Earth is given as approximately 5.96 × 10^24 kg, and its initial velocity is 0 m/s. Therefore, the initial momentum of the Earth is:
Momentum_earth = mass_earth × velocity_earth
= 5.96 × 10^24 kg × 0 m/s
= 0 kg·m/s
Since the total initial momentum of the system is the sum of the momenta of the person and the Earth, we have:
Initial momentum_system = Momentum_person + Momentum_earth
= 0 kg·m/s + 0 kg·m/s
= 0 kg·m/s
According to the conservation of momentum, the total momentum after the person jumps should also be zero, since there are no external forces acting on the system. Let's calculate the final momentum of the system to find the resulting speed of the Earth.
The mass of the person remains the same at 90 kg, and their velocity after jumping is given as 3.5 m/s. Therefore, the final momentum of the person is:
Momentum_person_final = mass_person × velocity_person_final
= 90 kg × 3.5 m/s
= 315 kg·m/s
Since the total final momentum is zero, the final momentum of the Earth should also be -315 kg·m/s to balance the system.
Now, let's calculate the final velocity of the Earth. We can rearrange the momentum formula to solve for velocity:
velocity_earth_final = momentum_earth_final / mass_earth
Plugging in the values we calculated, we get:
velocity_earth_final = (-315 kg·m/s) / (5.96 × 10^24 kg)
≈ -5.28 × 10^(-23) m/s
Therefore, the resulting speed of the Earth in the opposite direction is approximately -5.28 × 10^(-23) m/s.
Note: The negative sign indicates the direction opposite to the initial direction of the person's jump.