Dynamics Question: What is the velocity of a car traveling on a known curvature

[physicsnewblol]

Hi all,

I'm trying to write a Matlab simulation that determines the velocity of a car of known mass and moment of inertia which travels on a track whose curvature is also known.

To say the least, I'm at a loss as to what approach I should take to create my simulation. I'm finding it somewhat difficult to grasp the physics of the situation. The free body diagram is easy enough, but I can't recall how to couple this information with the constraint that the cart is confined to roll on the path.

If someone could provide some hints or a resource that I can read, I'd really appreciate it.


Thanks,

-PN

 

[physicsnewblol]

I've given this some thought, was wondering if some higher up could check my reasoning/math:

Coordinate system: x,y conventional.

Say we have a ramp given by the cubic function: y(x) = ax^{3} + bx^{2} + cx + d

If we start the cart at x = 0 and y = y_{0}, where y_{0}, is a maximum, the kinetic energy at any point is

Kinetic Energy T = \frac{1}{2}Mv_{x}(t)^{2} + \frac{1}{2}Iω^{2} = mg(y_{0} - y(t))

Assuming roll without slip condition: ω = \frac{v_{x}}{R} and some simplification we get:
T = Cv_{x}(t)^{2} = mg(y_{0} - y(t)) where C is some constant.

In order to get the equation solely in terms of x(t) and v_{x}(t), we can substitute y(t) for the cubic function y(x), which is implicitly a function of time through x.

By integrating both sides of the equation now, we can get a function for x(t), which we can plug back into y(x) to get y(t).Alright, how far off am I?
Thanks in advance,

-AN