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

1. Sep 12, 2006

### 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

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

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

$$$\begin{array}{l} \cos (\theta ) \ne \cos (2\theta ) \\ \sin (\theta ) \ne \sin (2\theta ) \\ 0 < \theta < \frac{\pi }{2} \\ \end{array}$$$

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. Sep 12, 2006

### 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.

3. Sep 12, 2006

### 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?

4. Sep 12, 2006

### 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 $\theta$ with the horizontal. What's its acceleration down the incline?