# A ball weighing 0.18 kg falls at a steady velocity v in a liquid with a density of 800 kg / m3.

A ball weighing 0.18 kg falls at a steady velocity v in a liquid with a density of 800 kg / m3. The volume of the ball is 100 cm3. With what force do you need to act on the ball so that it rises at a speed of 2v? Consider the drag force to be proportional to the speed.

m = 0.18 kg.
g = 9.8 m / s ^ 2.
ρ = 800 kg / m ^ 3.
V = 100 cm ^ 3 = 0.0001 m ^ 3.
Fcopr = k * v.
v2 = 2 * v1.
F -?
When the ball moves downward, 3 forces act on it: gravity m * g directed vertically downward, buoyancy force of Archimedes Farch = ρ * g * V directed vertically upward, resistance force to movement Fcopr1 = k * v1 directed upward.
m * g = ρ * g * V + k * v1.
k = (m * g – ρ * g * V) / v1.
When the ball moves upwards, 4 forces act on it: gravity force m * g directed vertically downward, buoyancy force of Archimedes Farch = ρ * g * V directed vertically upward, force of resistance to movement Fcopr2 = k * v2 directed downward, force F, s which the ball is lifted up.
m * g + k * v2 = ρ * g * V + F.
F = m * g + k * v2 – ρ * g * V.
Fcopr2 = k * v2 = 2 * k * v1 = 2 * (m * g – ρ * g * V) * v1 / v1 = 2 * (m * g – ρ * g * V).
F = m * g + 2 * (m * g – ρ * g * V) – ρ * g * V.
F = 0.18 kg * 9.8 m / s ^ 2 + 2 * (0.18 kg * 9.8 m / s ^ 2 – 800 kg / m ^ 3 * 9.8 m / s ^ 2 * 0 , 0001 m ^ 3) – 800 kg / m ^ 3 * 9.8 m / s ^ 2 * 0.0001 m ^ 3 = 2.94 N.
Answer: a force F = 2.94 N. must be applied to the ball.

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