- #1
opticaltempest
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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?
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?
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