Answer:
The answer to the question is
It would take about 167.021 s to splash down in the arctic ocean.
Step-by-step explanation:
The gravitational force is given by
Where m₁ mass of the piano and
m₂ = mass of the Earth
r = 3·R where R = radius of the earth
as stated in the question we have varying acceleration due to the inverse square law
Therefore at 3 × Radius of the earth we have
= 109.083 N and the acceleration =1.09083 m/s²
If the body falls from 3·R to 2·R with that acceleration we have
S = u·t +0.5×a·t² = 0.5×a·t² as u = 0
That is 6371 km = 0.5·1.09083·t²
t₁ = 108.079 s and we have
v₁² = u₁² +2·a₁·s₁ = 2·a₁·s₁ = 117.895 m/s
For the next stage r₂ = 2R
Therefore F = 245.436 N and a₂ = F/m₁ = 2.45436 m/s²
Therefore the time from 2R to R is given by
S₂ =R=u·t+0.5·a₂·t² = v₁·t + 0.5·a₂·t²
or 6371 km = 117.895 m/s × t + 0.5 × 2.45436 × t²
Which gives 1.22718 × t² + 117.895 × t -6371 = 0
Factorizing we have (t+134.631)(t-38.56)×1.22718 = 0
Therefore t = -134.631 s or 38.56 s as we only deal with positive values of time in the present question we have t₂ = 38.56 s and
v₂² = v₁² + 2·a₂·S = (117.895 m/s)² + 2·2.45436 m/s²×6371 km = 45172.686
v₂ = 212.54 m/s
For final stage we have r = R and
= 981.746 N and a₂ = F/m₁ = 9.81746 m/s²
Therefore the time from R to the arctic ocean is given by
S₃ =R=v₂·t+0.5·a₂·t² = 212.54·t + 0.5·9.81746·t² = 6371
Which gives
Therefore t₃ = 20.382 s
Therefore, it will take about t₁ + t₂ + t₃ = 108.079 s + 38.56 s + 20.382 s = 167.021 s to splash down in the ocean