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Homework Help: Parametric Equations Word Problem

  1. Jul 20, 2010 #1
    1. The problem statement, all variables and given/known data
    Consider a projectile launched at a height of h feet above the ground at an angle θ with the horizontal. If the initial velocity is v0 feet per second, the path of the projectile is modled by the parametric equations
    x=(v0cos θ)t and y=h + (v0 sin θ)t-16t2.

    The center-field fence in a ballpark is 10 feet high and 400 feet from home plate. The baseball is hit 4 feet above the ground. It leaves the bat at an angle of θ degrees with the horizontal at a speed of 100 miles per hour.

    Find the minimum angle required for the hit to be a home run

    2. Relevant equations

    3. The attempt at a solution

    So your basic equations are


    by the question when x=400, y>10 the ball will pass over the fence

    so if I solve for the angle θ when x=400 and y=10 the angle I get should be the minimum passable.



    sub that into the other equation


    I can get a common denominator and get it to


    but I'm not quite sure what to do after here to solve for θ.
    Last edited: Jul 20, 2010
  2. jcsd
  3. Jul 20, 2010 #2


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    Homework Helper

    Is it 3 or 4 in the 2nd equation? Your original problem says 4 feet.

    Assuming that it's 3:
    (I assume you meant to put a 10 in for y.) From here, try writing in terms of tan θ. I see a tan θ and a sec2 θ "hidden" in this equation, and you can use the pythagorean identity 1 + tan2 θ = sec2 θ. That way you'll end up with a quadratic in tan θ.

  4. Jul 20, 2010 #3
    Yeah, its suppose to be 3. sorry.

    so I'd have







    Expand, factor out negative


    Quadratic equation


    For the minus I get .3299 for the plus I get 3.031 the plus has way to high of an angle when I run an arctan on it and .3299 comes out as 18.27 degrees but this is wrong because I can check by plugging it back into the equation. and when x=400, y does not equal 10. It's close though. θ is suppose to equal "about 19.4 degrees" by the answer key.
  5. Jul 20, 2010 #4


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    Homework Helper

    You forgot to multiply the left side by 146.672 here. As it is, I wouldn't multiply both sides by 146.672 at all. I would simplify the coefficient of sec2 θ first:

    [tex]7 &= 400 \tan \theta - \frac{16(400)^2 \sec^2 \theta}{146.67^2}[/tex]

    [tex]7 &= 400 \tan \theta - \frac{14400}{121} \sec^2 \theta[/tex]

    (440/3 ≈ 146.67.)
    At this point, if you want, you could multiply both sides by 121 and then use the pythagorean identity.

  6. Jul 20, 2010 #5
    Ok, now I see how to do it. Thanks!
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