What do the values represent in this equation?

[jfnn]

Homework Statement

Hi everyone. I did an experiment in my house where I took four different geometric circles to represent a solid sphere, a spherical shell, a solid cylinder, and a hoop. I rolled all the objects down an incline of about 8 degrees, took video footage, uploaded into the tracker software for analysis. I creates position vs. time graphs for each object (x-t) and all the graphs were parabolic (object was accelerating down the incline).

After this I got four equations, all of the form

x = At^2 + Bt + c.

I am asked to determine what A is. In my assumption, I think A would be the linear acceleration of the object? Is my logic right, or am I wrong?

After I figure out what this value is, I am asked to say what the linear acceleration is, wouldn't it be the value of A?

After this, I need to check the validity of the four equations describing the linear motion of the objects used. All of the form of a = (some fraction depending on object used)gsin(theta).. However, if I use the value of A to check the valid of these equations, they are all not valid. So that makes me think that I interpreted the meaning of A wrong...

Any help would be suggested. Thank you.

Homework Equations

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x = At^2 + Bt + c.
a = (some fraction depending on object used)gsin(theta)

The Attempt at a Solution

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Mentioned above. Thank you in advance.

 

[Orodruin]

In my assumption, I think A would be the linear acceleration of the object? Is my logic right, or am I wrong?

How is acceleration defined? How does it relate to A?

Did you try just dropping something to check that you get the correct gravitational acceleration?

 

[jfnn]

How is acceleration defined? How does it relate to A?

Did you try just dropping something to check that you get the correct gravitational acceleration?

Acceleration is the change in the velocity over time. From the graph, which is a horizontal position vs time graph, the velocity would be the slope created form multiple tangent curves to that graph. Then the slope of that would be the acceleration. The acceleration is constant. Thus, I reasoned that the linear acceleration (correct me if I am wrong, the same as the horizontal acceleration) is equivalent to the value A that was generated from the computer software for my experiment?

 

[jfnn]

How is acceleration defined? How does it relate to A?

Did you try just dropping something to check that you get the correct gravitational acceleration?

Furthermore, I did previous experiments in my house involving the gravitational constant and it was always off by like +/- 1 depending on the experiment (I got this from the vertical position vs. time graph for previous experiments).

 

[jfnn]

Furthermore, I did previous experiments in my house involving the gravitational constant and it was always off by like +/- 1 depending on the experiment (I got this from the vertical position vs. time graph for previous experiments).

How is acceleration defined? How does it relate to A?

Did you try just dropping something to check that you get the correct gravitational acceleration?

Do you think my logic is right in defining the parameter A as the linear acceleration of the rolling object and that error was sufficient enough to cause variation in the equations?

 

[TSny]

Acceleration is defined as rate of change of velocity. You cannot assume (or define) the acceleration to be the constant A in your equation. But, as @Orodruin hinted, A is related in some way to the acceleration.

You might try a web search for "constant acceleration".

 

[Orodruin]

Please, one post at a time. Posting three consecutive messages within two minutes makes it seem as if you did not think through what you wanted to say and fragments your post.

Acceleration is the change in the velocity over time.

So how do we describe this mathematically? How do we describe the velocity mathematically given $x$ as a function of $t$? How do we then describe the acceleration mathematically if given $x$ as a function of $t$?

 

[jfnn]

Please, one post at a time. Posting three consecutive messages within two minutes makes it seem as if you did not think through what you wanted to say and fragments your post.
So how do we describe this mathematically? How do we describe the velocity mathematically given $x$ as a function of $t$? How do we then describe the acceleration mathematically if given $x$ as a function of $t$?

Thank you for your response, so sorry for multiple posts, ill keep that in mind.

So if I look at the kinematics equation describing the horizontal position of an object, I see the equation: x = 1/2at^2 + vt + x0

When comparing this to the parabolic equation that I got form my experimental data points on the tracker software, I get an equation of the form: x = 1/2At^2 + Bt + C.

It makes it more clear now relating the two equations, as they both describe the horizontal motion over time.

Therefore, the parameter A, which is the only quadratic term, must equal 1/2at^2 therefore if we set equal:

At^2 = 1/2at^2

A=1/2a then the parameter A is 1/2 the linear acceleration. So to find the linear acceleration, I must multiply the parameter A by 2? Does that make more sense?

The velocity is the change in position over time, which would be equivalent to the slope of my x-t graph. This value is what the value of parameter B is.

Does this work? Thank you for your repose to my question, it sparked my brain and made me think of this..

 

[Orodruin]

A=1/2a then the parameter A is 1/2 the linear acceleration. So to find the linear acceleration, I must multiply the parameter A by 2? Does that make more sense?

Yes, this is correct.

The velocity is the change in position over time, which would be equivalent to the slope of my x-t graph. This value is what the value of parameter B is.

This is only true at $t = 0$.

 

[jfnn]

Yes, this is correct.This is only true at $t = 0$.

Oh you are exactly right. The velocity of the object obviously changes with time due to the constant acceleration, so this value will change.

I appreciate all your help. Finally time to get this done!