In principle, anytime someone jumps up, the earth moves in the opposite direction. To see why we are unaware of this motion, calculate the recoil speed of the earth when a 65.0 kg person jumps upward at a speed of 1.70 m/s.
v=?
Find this from conservation of momentum
Me*v=65*1.70
To calculate the recoil speed of the Earth when a person jumps upward, we can use the principle of conservation of momentum. According to this principle, the total momentum of a system remains constant if no external forces are acting on it.
First, determine the initial momentum of the person and the Earth system before the jump. The momentum of an object is calculated by multiplying its mass by its velocity.
The initial momentum of the person is given by:
momentum_person_initial = (mass_person) * (velocity_person_initial)
where,
mass_person = 65.0 kg (mass of the person)
velocity_person_initial = 1.70 m/s (initial velocity of the person)
Next, since the person and the Earth are considered a system, the total momentum must be conserved. Therefore, the initial momentum of the Earth is equal in magnitude but opposite in direction to the initial momentum of the person:
momentum_earth_initial = -momentum_person_initial
To find the recoil speed of the Earth (v), we need to calculate the momentum of the Earth (momentum_earth_final) and divide it by the mass of the Earth (mass_earth).
momentum_earth_final = (mass_earth) * (velocity_earth_final)
Since the Earth is much more massive than the person, we can assume its recoil velocity (velocity_earth_final) is negligibly small compared to the person's initial velocity (velocity_person_initial). Thus, we can ignore the recoil velocity of the Earth (velocity_earth_final) when calculating the momentum of the Earth.
Therefore, the final momentum of the Earth is:
momentum_earth_final = 0
Using the conservation of momentum principle, we have:
momentum_earth_initial + momentum_earth_final = 0
Substituting the values, we get:
(-momentum_person_initial) + 0 = 0
Solving for momentum_person_initial, we find:
momentum_person_initial = 0
Now, we can solve for the recoil speed of the Earth (v) by dividing the initial momentum of the person by the mass of the Earth:
v = momentum_person_initial / mass_earth
The mass of the Earth is approximately 5.97 x 10^24 kg.
Substituting the values, we have:
v = 0 / (5.97 x 10^24 kg)
Therefore, the recoil speed of the Earth when a 65.0 kg person jumps upward at a speed of 1.70 m/s is negligible.