Does Increasing Ramp Angle Double Ball Speed?

[opticaltempest]

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

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

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

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

 

[Doc Al]

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.

 

[opticaltempest]

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?

 

[Doc Al]

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?

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

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?

Imagine a block sliding down a frictionless slope making an angle $\theta$ with the horizontal. What's its acceleration down the incline?