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

• physicsnewblol
In summary, the conversation discusses the creation of a Matlab simulation for determining the velocity of a car on a track with known mass, moment of inertia, and curvature. The participants are struggling with understanding the physics involved and request for hints or resources. The proposed approach involves using a ramp function to calculate kinetic energy and integrating to obtain a function for x(t).
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

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?

-AN

## 1. What is the meaning of "curvature" in this context?

The curvature in this context refers to the degree of curvature or bend in the path that the car is traveling on. It can also be thought of as the rate at which the direction of the car changes.

## 2. How is the velocity of a car affected by the curvature of the road?

The velocity of a car is affected by the curvature of the road in that it must change its direction in order to follow the curve. This change in direction requires a force, which can affect the speed of the car.

## 3. Is the velocity of a car traveling on a known curvature constant?

No, the velocity of a car traveling on a known curvature is not constant. As the car follows the curve, its velocity will change due to the force needed to change its direction. The velocity may also change if there are other factors such as friction or air resistance.

## 4. How does the velocity of a car change as the curvature of the road changes?

The velocity of a car will change as the curvature of the road changes. If the curvature increases, the car will need to turn more sharply and therefore its velocity will decrease. Conversely, if the curvature decreases, the car will need to turn less sharply and its velocity may increase.

## 5. Can the velocity of a car traveling on a known curvature be calculated?

Yes, the velocity of a car traveling on a known curvature can be calculated using the formula v = sqrt(r*a), where v is the velocity, r is the radius of the curve, and a is the acceleration. However, this calculation may not account for all factors such as friction and air resistance, so the actual velocity may differ slightly.

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