Does Increasing Ramp Angle Double Ball Speed?

In summary, the ball does not roll down at double the speed when the angle of the ramp is increased by a factor of two.
  • #1
opticaltempest
135
0
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:
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  • #2
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
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
opticaltempest said:
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 [itex]\theta[/itex] with the horizontal. What's its acceleration down the incline?
 

1. What is the relationship between ramp angle and ball speed?

The relationship between ramp angle and ball speed is directly proportional. This means that as the ramp angle increases, the ball speed also increases. As the ramp angle decreases, the ball speed decreases.

2. How does increasing ramp angle affect the trajectory of the ball?

Increasing ramp angle causes the trajectory of the ball to become steeper. This is because the ball is launched at a higher angle, resulting in a sharper trajectory.

3. Does the weight of the ball impact the results of increasing ramp angle?

Yes, the weight of the ball can impact the results of increasing ramp angle. A heavier ball will require more force to be launched at the same angle, resulting in a slower ball speed compared to a lighter ball.

4. Are there any limitations to the increase in ramp angle and its impact on ball speed?

Yes, there are limitations to the increase in ramp angle and its impact on ball speed. Eventually, the ball will reach a maximum speed and any further increase in the ramp angle will not result in a significant increase in ball speed.

5. How can I conduct an experiment to test the relationship between ramp angle and ball speed?

To conduct an experiment, you will need a ramp, a ball, a measuring device, and a way to change the ramp angle. Start by setting the ramp at a low angle and launch the ball. Measure the speed of the ball using the measuring device. Then, increase the ramp angle and repeat the process. Record the results and analyze the data to observe the relationship between ramp angle and ball speed.

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