Anonymous
Anonymous asked in Social ScienceOther - Social Science · 1 month ago

Doing this problem?

If you're playing volleyball and one of the teammates serves the ball by hitting it from a height of 8 ft in arc that is parabolic, imagine that the ball reaches its max height of 12 ft, 4 ft to the right of where it was originally served. But after traveling 4 more feet after that, it gets stuck on the net. 

1) What is the equation representing the trajectory of the volleyball if x and y represent horizontal and vertical distance from the start point?

2) How high is the net?

2 Answers

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  • 1 month ago

    The physical method would be to set up equations involving gravity and time.

    However, we can model on a quadratic function that relates x to y

    so, y = -a(x - b)² + c 

    As this is in vertex form and we know (4, 12) gives the maximum value we have:

    y = -a(x - 4)² + 12

    Also, when x = 0, y = 8 so,

    8 = -a(-4)² + 12

    i.e. 8 = -16a + 12

    a = 1/4

    Hence, y = (-1/4)(x - 4)² + 12

    When x = 4, the ball reaches the greatest height of 12 which is midway along it's horizontal flight.

    Hence, if it travels another 4 feet, due to symmetry, it returns to the vertical level of projection.

    so, height of net is 8 feet

    Note, this is the same as saying, what is y when x = 8

    => y = (-1/4)(8 - 4)² + 12

    i.e. y = (-1/4)(16) + 12 => 8

    Going a bit further, it will hit the ground when:

    0 = (-1/4)(x - 4)² + 12

    Giving, x = 11.9 feet

    A sketch is below.

    :)>

    Attachment image
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  • Using calculus:

    x(t) = d/dx v(t) = d²/dx² a(t)

    a(t) = -9.8

    v(t) = -9.8t + C → C represents the initial velocity

    x(t) = -4.9t² + Ct + D → D represents the initial height (8 ft)

    Interpreting the max height:

    "...reaches its max height of 12 ft..."

    v(t) = 0 = -9.8t + C

    t = C / 9.8

    x(C/9.8) = 12 = -4.9(C/9.8)² + C(C/9.8) + 8

    ... Solve for C 

    x(t) = -4.9t² + Ct + 8 → Insert C to get your function

    ___

    To find the height of the net, we know it travels 4 feet to the right after that. Since parabolas are symmetric, and the max height occurs 4 feet right of where it was served, just plug in:

    x(2C/9.8) = -4.9(2C/9.8)² + C(2C/9.8) + 8

    And you solved for C previously

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