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13 October, 00:47

Use the position function s (t) = - 16t + v_0t + s_0 for free falling objects. A ball is thrown straight down from the top of a 600-foot building with an initial velocity of - 30 feet per second. (a) Determine the position and velocity functions for the ball. (b) Determine the average velocity on the interval [1, 3]. (c) Find the instantaneous velocities when t=1 and t=3. (d) Find the time required for the ball to reach ground level. (e) Find the velocity of the ball at impact.

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  1. 13 October, 01:02
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    a) v = - 30 - 32 t, s (t) = 600 - 30 t - 16 t², b) v = - 32 ft / s

    c) v (1) = - 62 ft / s, v (3) = - 126 ft / s, d) t = 7.13 s, e) v = - 258.16 ft / s

    Explanation:

    a) For this exercise they give us the function of the position of the ball

    s (t) = s (o) + v_o t - 16 t²

    notice that you forgot to write the super index

    indicate the initial position of the ball

    s (o) = 600 ft

    also indicates initial speed

    v_o = - 30 ft / s

    let's substitute in the equation

    s (t) = 600 - 30 t - 16 t²

    to find the speed we use

    v = ds / dt

    v = v_o - 32 t

    v = - 30 - 32 t

    b) To find the average speed, look for the speed at the beginning and end of the time interval

    t = 1 s

    v (1) = - 30 - 32 1

    v (1) = - 62 ft / s

    t = 3 s

    v (3) = - 30 - 32 3

    v (3) = - 126 ft / s

    the average speed is

    v = (v (3) - v (1)) / (3-1)

    v = (-126 + 62) / 2

    v = - 32 ft / s

    c) instantaneous speeds, we already calculated them

    v (1) = - 62 ft / s

    v (3) = - 126 ft / s

    d) the time to reach the ground

    in this case s = 0

    0 = 600 - 30 t - 16 t²

    t² + 1,875 t - 37.5 = 0

    we solve the quadratic equation

    t = [-1,875 ±√ (1,875² + 4 37.5) ] / 2

    t = [1,875 ± 12.39] / 2

    t₁ = 7.13 s

    t₂ = negative

    Since the time must be positive, the correct answer is t = 7.13 s

    e) the speed of the ball on reaching the ground

    v = - 30 - 32 t

    v = - 30 - 32 7.13

    v = - 258.16 ft / s
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