Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The range of a projectile fired up the side of a mountain?

  1. Aug 4, 2004 #1

    cj

    User Avatar

    Another question pertaining to a lab that I have to setup...

    The range of a projectile fired up the side of a mountain?

    At the base of a mountain with a constant slope of
    [tex]\phi[/tex], a projectile is fired from a cannon oriented
    at an angle of [tex]\theta[/tex].

    What [tex]\theta[/tex] will result in the maximum range up the
    side of the mountain?

    What is this maximum range??

    - air resistance can be neglected.

    ---
    As a starting point I'm looking at

    [tex]x_{max}=\frac{v_0^2sin2\alpha}{g}[/tex]

    But am not clear about next steps.
     
  2. jcsd
  3. Aug 4, 2004 #2

    NateTG

    User Avatar
    Science Advisor
    Homework Helper

    This is a classic problem.

    That said, it's not at all clear that your formula applies to this situation. Perhaps it's easier to calculate the distance the projectile travels in terms of [tex]\phi[/tex],[tex]\theta[/tex] and [tex]v_0[/tex] and then find the maximum?
     
  4. Aug 4, 2004 #3

    cj

    User Avatar

    Yes, this is the line of thinking I was considering -- but came to a dead-end.

     
  5. Aug 4, 2004 #4

    arildno

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Find the equation for the parabola with the correct slope (y as a function of x).
    Find the intersection of this parabola with the straight line representing the mountain slope (i.e, find the x-coordinate of the point of intersection).

    alternatively, write the equations of motion in terms of normal and tangential components to the mountainside. Determine the time at which the projectile's normal position is again 0.

    Anyhow, once you've found this, you may find an equation for distance as a function of the angle that can be maximized.
     
    Last edited: Aug 4, 2004
  6. Aug 4, 2004 #5
    Form a triangle so that you have the slope representing the distance travelled up the mountain.
    (u cos theta) t is horizontal distance (theta = angle between initial velocity vector,u, and horizontal) and 1/2 gt ^ 2 is vertical distance on triangle.

    distance up mountain = sqrt [ (1/2gt ^2)^ 2 + (u cos theta)^2 t ^ 2]

    Since distance up mountain is a function of time:
    for maximum distance up mountain

    d(distance)/ dt = 0

    so d/dt sqrt [ (1/2gt ^2)^ 2 + (u cos theta)^2 t ^ 2]= 0

    If phi is angle of mountain slope, then
    tan phi = vertical of triangle/ horizontal of triangle
    tan phi = 1/2 gt ^ 2 / ( u cos theta) t
    so 1/2 gt^2 = (u cos theta) t x tan phi
    u cos theta = (1/2 gt)/tan phi

    so d/dt sqrt [ (1/2gt ^2)^ 2 + (1/2 gt) ^2/ (tan phi)^2 t ^ 2]= 0
    d/dt sqrt [ 1/4 g^2 x t^4 + 1/4g^2 x t^2/ (tan phi)^2 t^2 ]= 0

    for constant phi we can evaluate d/dt.


    ( differentiation can be done using u substitution and chain rule)

    substituting the theta and phi for t in the differentiated expression
    will give maximum distance projectile travels up mountain.
     
    Last edited: Aug 5, 2004
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: The range of a projectile fired up the side of a mountain?
  1. Projectile motion range (Replies: 10)

Loading...