A sphere made of plastic has a density of 1.36 g/cm3 and a radius of 8.50 cm. It falls through air of density 1.20 kg/m3 and has a drag coef

Question

A sphere made of plastic has a density of 1.36 g/cm3 and a radius of 8.50 cm. It falls through air of density 1.20 kg/m3 and has a drag coefficient of 0.500. What is its terminal speed (in m/s)? From what height (in m) would the sphere have to be dropped to reach this speed if it fell without air resistance?

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Huyền Thanh 4 years 2021-08-30T15:49:09+00:00 1 Answers 10 views 0

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  1. thuthao
    0
    2021-08-30T15:51:05+00:00
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    Answer:

    The answers to the questions are

    Part 1

    The terminal speed of the sphere (in m/s) is 44.8973 m/s.

    Part 2

    The sphere has to be dropped from a height of 102.74 m to reach the speed of 44.8973 m/s if it fell without air resistance.

    (102.74 m).

    Explanation:

    Part 1

    Terminal velocity is given by

    v_T = \sqrt{\frac{mg}{\rho C_DA} }

    Where

    v_T = Terminal velocity

    m = Mass of the sphere = 3.498 kg

    g = Acceleration due to gravity = 9.81 m/s²

    ρ = Density of the air = 1.20 kg/m³

    C_D = Drag coefficient = 0.500

    A = Object cross sectional area = 0.022698 m²

    The mass of the sphere is given by

    Density = \frac{Mass}{Volume} where volume of the sphere is given by

    Volume = \frac{4}{3} \pi r^3

    Where

    r = radius of the sphere = 8.50 cm = ‪0.085‬ m

    Therefore the volume V = 2572.44 cm³ = 2.57×10⁻³ m³

    The mass then becomes, Mass = Density × Volume  = 1.36×2572.44 = 3498.5184 g = 3.498 kg

    The mass, m of the sphere = 3.498 kg

    The cross sectional area of the sphere is given by A = π×r² = 0.022698 m²

    Therefore the terminal velocity becomes

    v_T = \sqrt{\frac{3.498*9.81}{1.20*0.500*0.022698} }

    = 44.8973 m/s.

    Part 2

    To reach the speed of 44.8973 m/s in free fall, we have

    v² = u² + 2·g·h where

    u = initial velocity = 0 m/s

    h = Height from which the sphere needs to be dropped

    v = final sphere velocity without air resistance

    Therefore v² = 2·g·h ⇒ 44.8973² = 2×9.81×h

    h = 44.8973²÷(2×9.81)

    = 102.74 m.

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