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14 May, 03:58

A bolt is dropped from a bridge under construction, falling 96 m to the valley below the bridge. (a) How much time does it take to pass through the last 11 % of its fall? What is its speed (b) when it begins that last 11 % of its fall and (c) just before it reaches the ground?

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  1. 14 May, 07:19
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    a) It takes the bolt 0.25 s to pass the last 11% of the fall.

    b) When the bolt begins to fall the last 11% of the fall its velocity is - 41.2 m/s.

    c) The velocity of the bolt just before it reaches the ground is - 43.6 m/s

    Explanation:

    Hi there!

    a) Let's calculate how much distance it is the last 11% of the fall:

    96 m · 0.11 = 10.56 m

    So, we have to find how much time it takes the bolt to pass from a height of 10.56 m to the ground.

    First, let's calculate how much time it takes the bolt to reach a height of 10.56 m. For that we can use this equation:

    h = h0 + v0 · t + 1/2 · g · t²

    Where:

    h = height of the bolt at a time t.

    h0 = initial height.

    v0 = initial velocity.

    t = time.

    g = acceleration due to gravity.

    If we consider the ground as the origin of the frame of reference, then h0 = 96 m. Since the bolt is dropped, the initial velocity is zero (v0 = 0). Then, the equation gets reduce to this:

    h = h0 + 1/2 · g · t²

    We have to find at which time h = 10.56 m.

    10.56 m = 96 m - 1/2 · 9.8 m/s² · t²

    Solving for t:

    √ (-2 · (10.56 m - 96 m) / 9.8 m/s²) = t

    t = 4.2 s

    Now that we have the time at which the bolt is located at 10.56 m above the ground, we can calculate the velocity of the bolt at that time.

    The equation of velocity (v) of the bolt is the following:

    v = v0 + g · t

    at t = 4.2 s.

    v = 0 - 9.8 m/s² · 4.2 s

    v = - 41.2 m/s

    When the bolt begins to fall the last 11% of the fall its velocity is - 41.2 m/s.

    Now, we can calculate how much time it takes to fall the last 10.56 m.

    The initial velocity of the bolt will be the velocity at h = 10.56 m. The initial height will be 10.56 m.

    h = h0 + v0 · t + 1/2 · g · t²

    We have to find the time at which h = 0 (the bolt hits the ground)

    0 = 10.56 m - 41.2 m/s · t - 1/2 · 9.8 m/s² · t²

    Solving the quadratic equation using the quadratic formula:

    t = 0.25 s (the other solution of the quadratic equation is negative and thus discarded).

    It takes the bolt 0.25 s to pass the last 11% of the fall.

    Now, let's calculate the velocity of the bolt when it reaches the ground:

    v = v0 + g · t

    v = - 41.2 m/s - 9.8 m/s² · 0.25 s

    v = - 43.6 m/s

    The velocity of the bolt just before it reaches the bolt is - 43.6 m/s
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