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: $y(x)$ = a$x^{3}$ + b$x^{2}$ + $cx$ + 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 = $\frac{1}{2}$M$v_{x}(t)^{2}$ + $\frac{1}{2}$I$ω^{2}$ = mg($y_{0}$ - $y(t)$)

Assuming roll without slip condition: ω = $\frac{v_{x}}{R}$ and some simplification we get:
T = C$v_{x}(t)^{2}$ = 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