1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Derive optimal anlge for maximum range (Projectile Motion)

  1. May 3, 2012 #1
    1. The problem statement, all variables and given/known data
    Hey,

    For an assignment I need to derive the optimal angle for maximum range (derive Equation 2 below). I know how to derive equation 1 but I need to derive the second equation so I can substitute h = 0, and h = 1, into it, to show the optimal angle for maximum range for bi-level and uni-level.

    I have a book that explains it but I don't understand it at all. Any help is really appreciated.

    Thank you!

    2. Relevant equations
    Equation 1:
    Rmax = u^2 / g √(1 + 2gh/u^2)

    Equation 2:
    tan θ = 1 / √ (1 + 2gh / u^2)

    u = initial velocity
    g = grav. accel.
    h = height

    3. The attempt at a solution
    One method I have tried is finding the derivative of Rmax, setting it to zero and solving for theta (to find the minimum) which does give me 45 degrees but I cannot apply this to bi-level projection.
     
  2. jcsd
  3. May 4, 2012 #2
    The way you've written the fractions is somewhat ambiguous. It's hard to say whether you meant that [itex]R_{\text{max}}=\frac{u^2}{g}\frac{1}{\sqrt{1+\frac{2gh}{u^2}}}[/itex] or [itex]R_{\text{max}}=\frac{u^2}{g}\sqrt{1+\frac{2gh}{u^2}}[/itex] (especially since the term in the radical is dimensionless), but I've realized that it's the second one - the moral of the story is to use parentheses :smile:.

    In this case, one method is to use the standard [itex]x(t)[/itex] and [itex]y(t)[/itex] to find [itex]y(x)[/itex] and sub in [itex]y(R)=0[/itex] to get an equation that defines [itex]R[/itex] implicitly as a function of [itex]\theta[/itex]. Knowing this, you can find [itex]\tan(\theta_{\text{max}})[/itex] as a function of [itex]R_{\text{max}}[/itex], which you said you already found.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Derive optimal anlge for maximum range (Projectile Motion)
Loading...