- #1

Atomillo

- 34

- 3

- Homework Statement
- Let f(x) be the shape of a line. A point is located in some initial position of this line. What are the equations of motion of this point?

- Relevant Equations
- Conservation of Energy.

Hi!

First of all, mention that this is not a "homework" problem in the sense that no teacher ever gave it to me or that I have the obligation to do it. It is a question that came to mind when repasing the theory done in class and though interesting. I still post it here because I suppose that what I'm missing is something basic, but in case it didn't belong to this category please tell me so that I can delete it and comply with the rules. Thanks in advance.

So, I'm currently in first semester of Electronic Engineering after doing HL Physics in IB. In class we studied the motion of a point in an inclined plane. I wonder however how could we find the equations of motion for a point in a line (2D, to keep things as simple as possible) that is not straight, i.e that is defined by some function f(x).

I naturally came to the conservation of energy theorem. After some algebraic manipulation to find the velocity, I got:

This is my first post, so if I'm doing something incorrectly, posting on the wrong place, etc... please tell me so that I can correct it as soon as possible.

Thanks!

EDIT: I said that this was my first post. This is not true! It has been so long since I posted (lurking only) that I completely forgot. My apologies.

First of all, mention that this is not a "homework" problem in the sense that no teacher ever gave it to me or that I have the obligation to do it. It is a question that came to mind when repasing the theory done in class and though interesting. I still post it here because I suppose that what I'm missing is something basic, but in case it didn't belong to this category please tell me so that I can delete it and comply with the rules. Thanks in advance.

So, I'm currently in first semester of Electronic Engineering after doing HL Physics in IB. In class we studied the motion of a point in an inclined plane. I wonder however how could we find the equations of motion for a point in a line (2D, to keep things as simple as possible) that is not straight, i.e that is defined by some function f(x).

I naturally came to the conservation of energy theorem. After some algebraic manipulation to find the velocity, I got:

v(h) = √(2*(E-mgh)/m)

Where E is the total mechanical energy at the starting point and mgh is the potential energy at a height h. However, if the shape of the plane is expressed in the function f(x) as stated previously, then the height h at a point a is simply f(a). Re writing:

v(x) = √(2*(E-mgf(x))/m)

Here I came to my first problem: I was looking for v(t) and got v(x). Perusing the internet, I found this very helpful thread which I followed: Time dependence of velocity from position dependence of velocity. After doing as stated in the forum, I got:

t + C = ∫1/(v(x)) dx

After solving the integral, the only thing left to do would be to solve for x. In order to check that this result is correct, I then try to derive from this general case the equations of motion for an inclined plane of 45 degrees. Now, this plane is characterized by the equation f(x) = -x. Replacing, I get:

v(x) = √(2*(E+mgx)/m)

Using an HP Prime I solved the integral to obtain:

t + C = √2 * √(gm^2x+mE)/gm

Assuming C = 0, x is then:

x(t) = (g^2mt^2-2E)/2gm

Now, assuming the point starts at x = 0, that the initial velocity is cero, then E = 0 (since E is the initial mechanic energy, and starting at x=0 that means h=0 and there is no kinetic energy). After all this, I get:

x(t) = (g*t^2)/2

This doesn't make any sense! This equation is describing the motion of free fall i.e an angle of 90 degrees! How could this happen? Does this prove that the third equation is incorrect, or did I screw something up in the way to the result?

This is my first post, so if I'm doing something incorrectly, posting on the wrong place, etc... please tell me so that I can correct it as soon as possible.

Thanks!

EDIT: I said that this was my first post. This is not true! It has been so long since I posted (lurking only) that I completely forgot. My apologies.