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24 April, 14:01

An object of mass 100 kg is released from rest from a boat into the water and allowed to sink. While gravity is pulling the object down, a buoyancy force of StartFraction 1 Over 50 EndFraction times the weight of the object is pushing the object up (weight = mg). If we assume that water resistance exerts a force on the object that is proportional to the velocity of the object, with proportionality constant 10 N-sec/m, find the equation of motion of the object. After how many seconds will the velocity of the object be 90 m/sec? Assume that the acceleration due to gravity is 9.81 m divided by sec squared.

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  1. 24 April, 17:57
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    a) v_f = 9.6138*t / (1 + 0.1*t)

    b) t = 147 s

    Step-by-step explanation:

    Given:

    - The mass of the object m = 100 kg

    - The buoyancy force F_b = W / 50

    - The weight of the object = W = m*g

    - Water resistance F_w = k*v

    - Where, k = 10 Ns/m

    Find:

    a) find the equation of motion of the object.

    b) After how many seconds will the velocity of the object be 90 m/sec?

    Solution:

    - Construct a FBD of the object, three forces acting on the object. Weight downwards, Buoyancy and drag force upwards. Use Newton's Second law of motion to evaluate the acceleration a.

    F_net = m*a

    W - W/50 - 10*v = m*a

    a = 49/50*g - 10/m * v

    Plug in the values:

    a = 49/50 * 9.81 - 10/100 * v

    a = 9.6138 - 0.1*v

    - Using the first kinematic equation of motion we have:

    v_f = v_i + a*t

    object was dropped from rest, v_i = 0, Hence:

    v_f = (9.6138 - 0.1*v) * t

    v_f = 9.6138*t / (1 + 0.1*t)

    Find the time t when v_f = 90 m/s

    90 (1 + 0.1*t) = 9.6138*t

    90 + 9*t = 9.6138*t

    t = 90 / 0.6138 = 147 s
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