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opticaltempest
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A ball is rolled down a frictionless incline at some angle \theta 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
\[<br /> \begin{array}{l}<br /> Speed = \sqrt {\left( {v_i \cos \left( \theta \right)t} \right)^2 + \left( {h + v_i \sin \left( \theta \right)t + \frac{1}{2}gt^2 } \right)^2 } \\ <br /> 2Speed \ne \sqrt {\left( {v_i \cos \left( {2\theta } \right)t} \right)^2 + \left( {h + v_i \sin \left( {2\theta } \right)t + \frac{1}{2}gt^2 } \right)^2 } \\ <br /> \end{array}<br /> \]<br />
Where v_i is the initial velocity of the ball, \theta is the angle of the ramp below the horizontal, h is the initial height of the ball, g is the acceleration due to gravity, and t time.because
<br /> \[<br /> \begin{array}{l}<br /> \cos (\theta ) \ne \cos (2\theta ) \\ <br /> \sin (\theta ) \ne \sin (2\theta ) \\ <br /> 0 < \theta < \frac{\pi }{2} \\ <br /> \end{array}<br /> \]<br />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?
Here is what I said:
No. because of the following relation
\[<br /> \begin{array}{l}<br /> Speed = \sqrt {\left( {v_i \cos \left( \theta \right)t} \right)^2 + \left( {h + v_i \sin \left( \theta \right)t + \frac{1}{2}gt^2 } \right)^2 } \\ <br /> 2Speed \ne \sqrt {\left( {v_i \cos \left( {2\theta } \right)t} \right)^2 + \left( {h + v_i \sin \left( {2\theta } \right)t + \frac{1}{2}gt^2 } \right)^2 } \\ <br /> \end{array}<br /> \]<br />
Where v_i is the initial velocity of the ball, \theta is the angle of the ramp below the horizontal, h is the initial height of the ball, g is the acceleration due to gravity, and t time.because
<br /> \[<br /> \begin{array}{l}<br /> \cos (\theta ) \ne \cos (2\theta ) \\ <br /> \sin (\theta ) \ne \sin (2\theta ) \\ <br /> 0 < \theta < \frac{\pi }{2} \\ <br /> \end{array}<br /> \]<br />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?
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