The relationship between Vav and Height of a ball rolling down a slope

[zzoldan]

Homework Statement

I have this problem on my physics exam and I hoped you guys could help. The Question is: Why is the relationship between the average velocity and the height of a ball rolling down a slope a square root function?
The slope is 100cm long, the height of the ball at the top of the slope is 30cm and the angle is 17.45 degrees.

Homework Equations

S=1/2at2 + viT
Vav = S/T
Vav = (Vi + Vf) / 2

The Attempt at a Solution

I established that the relation is, in fact, represented by a square root function.
I figured out the rule of the function, by determining that the A/K is.
I just have no idea why this is a square root function...
Is it because the ball reaches its' terminal velocity quicker?
Or because, when the height is smaller, there is a much smaller acceleration, ultimately decreasing the average velocity?
Please Help- My exam is on 12/17/09

Thanks :D

 

[Nate810]

The relationship between the average velocity and the height of a ball rolling down a slope is a square root function because the acceleration of the ball due to gravity decreases as it gets closer to the bottom of the slope. As the ball moves down the slope, its kinetic energy (KE) is converted to potential energy (PE) due to the force of gravity acting upon it. As it gets closer to the bottom of the slope, the PE increases while the KE decreases, resulting in a decrease in the acceleration. This decrease in acceleration results in a decrease in the average velocity of the ball, which can be represented by a square root function.

 

[kerimek]

I would approach this problem by first understanding the physical principles involved. When a ball rolls down a slope, it experiences both gravitational and frictional forces. The gravitational force accelerates the ball down the slope, while the frictional force acts in the opposite direction to slow it down. As the ball moves down the slope, its velocity increases due to the acceleration from gravity, but at the same time, the frictional force also increases, eventually balancing out the gravitational force and causing the ball to reach a constant velocity known as the terminal velocity.

Now, let's consider the relationship between the average velocity and the height of the ball. As the ball rolls down the slope, it covers a certain distance in a specific amount of time, giving us its average velocity. Since the ball starts from rest at the top of the slope, its initial velocity is 0. Therefore, the average velocity is equal to the final velocity divided by 2. Now, let's look at the equation for the final velocity of the ball:

Vf2 = Vi2 + 2ad

Where Vf is the final velocity, Vi is the initial velocity, a is the acceleration due to gravity, and d is the distance covered by the ball. We can rearrange this equation to solve for the distance d:

d = (Vf2 - Vi2) / 2a

Substituting this into the equation for average velocity, we get:

Vav = (Vi + Vf) / 2 = (Vi + √(Vi2 + 2ad)) / 2

Now, we can see that the average velocity is dependent on the initial velocity, which is 0, and the distance covered by the ball, which is directly related to the height of the slope. As the height of the slope increases, the distance covered by the ball also increases, resulting in a higher average velocity. However, as the ball reaches its terminal velocity, the distance covered by the ball will not increase any further, resulting in a constant average velocity.

So, to answer the question, the relationship between average velocity and height of a ball rolling down a slope is a square root function because the distance covered by the ball, and therefore the average velocity, is directly proportional to the square root of the height of the slope. As the height increases, the distance covered and the average velocity also increase, but at a slower rate due to the square root function. This is