1. Not finding help here? Sign up for a free 30min 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!

Ball rolling down frictionless incline - Does doubling the angle double the speed?

  1. Sep 12, 2006 #1
    A ball is rolled down a frictionless incline at some angle [tex]\theta[/tex] below the horizontal. If you increase the angle of the incline by a factor of two (make the ramp steeper downward by twice as much), does the ball roll down at double the speed?

    Here is what I said:

    No. because of the following relation

    [tex] \[
    \begin{array}{l}
    Speed = \sqrt {\left( {v_i \cos \left( \theta \right)t} \right)^2 + \left( {h + v_i \sin \left( \theta \right)t + \frac{1}{2}gt^2 } \right)^2 } \\
    2Speed \ne \sqrt {\left( {v_i \cos \left( {2\theta } \right)t} \right)^2 + \left( {h + v_i \sin \left( {2\theta } \right)t + \frac{1}{2}gt^2 } \right)^2 } \\
    \end{array}
    \]
    [/tex]

    Where [tex]v_i[/tex] is the initial velocity of the ball, [tex]\theta[/tex] is the angle of the ramp below the horizontal, [tex]h[/tex] is the initial height of the ball, [tex]g[/tex] is the acceleration due to gravity, and [tex]t[/tex] time.


    because

    [tex]
    \[
    \begin{array}{l}
    \cos (\theta ) \ne \cos (2\theta ) \\
    \sin (\theta ) \ne \sin (2\theta ) \\
    0 < \theta < \frac{\pi }{2} \\
    \end{array}
    \]
    [/tex]


    Is this a correct way to show that increasing the downward angle by a factor of two does not double the speed of the object rolling down the incline?
     
    Last edited: Sep 12, 2006
  2. jcsd
  3. Sep 12, 2006 #2

    Doc Al

    User Avatar

    Staff: Mentor

    Why not just write an expression for the acceleration along the incline as a function of angle? (Are you concerned with something rolling, or did you just mean sliding without friciton?)

    I assume you mean speed as a function of time as it goes down the incline. The speed at the bottom will only depend on the height.
     
  4. Sep 12, 2006 #3
    It is speed as a function of time, and I guess the object could be rolling or sliding with no resistance.

    Even though my answer isn't the simpliest, is it still a correct way to show that increasing the downward angle by a factor of two does not double the speed?

    What is the simplest way to show that the increasing the downward slope of the ramp by a factor of two does not double the speed?
     
  5. Sep 12, 2006 #4

    Doc Al

    User Avatar

    Staff: Mentor

    I don't understand your equation, since you have speed on one side but distance on the other.

    Imagine a block sliding down a frictionless slope making an angle [itex]\theta[/itex] with the horizontal. What's its acceleration down the incline?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Ball rolling down frictionless incline - Does doubling the angle double the speed?
Loading...