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Math Help - Antiderivative.

  1. #1
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    Antiderivative.

    A stone is dropped from CT Tower, 450m above the ground.
    If the stone is thrown downward with a speed of 5m/s,
    How long does it take to hit the ground...

    I thought the answer is so simple just ,,,450/5=90s but it is not..

    Please show me how to get t=9.09s
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  2. #2
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    Quote Originally Posted by camherokid View Post
    A stone is dropped from CT Tower, 450m above the ground.
    If the stone is thrown downward with a speed of 5m/s,
    How long does it take to hit the ground...

    I thought the answer is so simple just ,,,450/5=90s but it is not..

    Please show me how to get t=9.09s
    The equation is,

    h(t)=-16t^2+5t+450

    Solve,

    -16t^2+5t+450=0
    For positive root.
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  3. #3
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    I think the equation should be like this:

    h(t)=-4.9t^2-5t+450

    because I got the answer that I want.

    I appreciate that Mr. PerfectHacker
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  4. #4
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by camherokid View Post
    A stone is dropped from CT Tower, 450m above the ground.
    If the stone is thrown downward with a speed of 5m/s,
    How long does it take to hit the ground...

    I thought the answer is so simple just ,,,450/5=90s but it is not..

    Please show me how to get t=9.09s
    I think i got a different answer from TPH. I think that's because he worked in feet when the question has measurements in meters. here are the steps:

    the acceleration due to gravity in meters is 9.8 m/s^2, so our acceleration function is:

    a(t) = -9.8 ........since we are falling (moving in the negative direction)
    now our velocity function is v(t) = int{a(t)}dt = int{-9.8}dt = -9.8t + C
    now the initial velocity is -5, so
    v(0) = -9.8(0) + C = -5
    => C = -5
    so v(t) = -9.8t - 5

    our position function is given by
    s(t) = int{v(t)}dt = int{-9.8t - 5}dt = -4.9t^2 - 5t + C
    our initial position is 450 m above ground, so
    s(0) = -4.9(0)^2 - 5(0) + C = 450
    => C = 450
    so s(t) = -4.9t^2 - 5t + 450

    we want t when s(t) = 0
    => -4.9t^2 - 5t + 450 = 0
    By the quadratic formula:
    t = [5 +/- sqrt(25 + 8820)]/-9.8
    => t = -10.11s or t = 9.09s
    t cannot be negative, so t = 9.09s
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  5. #5
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by camherokid View Post
    I think the equation should be like this:

    h(t)=-4.9t^2+5t+450

    because I got the answer that I want.

    I appreciate that Mr. PerfectHacker
    you won't get the answer you want with that equation, the 5t should be negative, otherwise, you'll get t = -9.09 and 10.11 and our answer should be +9.09 as you said
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  6. #6
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    Hello, camherokid!

    A stone is dropped from CT Tower, 450m above the ground.
    If the stone is thrown downward with a speed of 5m/s,
    how long does it take to hit the ground?
    The general (metric) formula is: .h .= .ho + vot - 4.9t

    . . where h
    o is the initial height and vo is the initial velocity.


    We are given: .h
    o = 450, vo = -5

    The function is: .h .= .450 - 5t - 4.9t


    Solve h = 0: .4.9t + 5t - 450 .= .0
    . . . . . . . . . . . . . . . . . . . . . . .______________
    . . . . . . . . . . . . . . . . . . -5 √5 - 4(4.9)(-450)
    Quadratic Formula: . t .= .---------------------------- . .9.09 seconds
    . . . . . . . . . . . . . . . . . . . . . . . . 2(4.9)

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  7. #7
    Forum Admin topsquark's Avatar
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    Just a quick comment as to WHY the 5 should be negative.

    All of you are working in a coordinate system where the ground is the origin and +y is upward.

    The initial position (450 m) is positive because it is above the origin and + is upward. The intial velocity is negative because the ball is being thrown downward. And the acceleration is negative because the weight acts in a downward direction.

    ALWAYS choose and label a coordinate system!!

    -Dan
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