The Time of Sliding Points Down a Curve and Parametric Equations

[Caesius]

I am taking Calculus III for one of my classes and Im learning about solving for Tangent Vectors and Normal Vectors for a curve. I figured out a way to determine the g-force exerted on a "particle" at a particular point on the curve by using vector calculus.

So, just for laughs, I wanted to find out what g-forces a particle would experiance on the curve f(x) = x². However, in order to solve it, I need the parametric equations for f(x) = x². Im trying to make the parametric equations work so f(x) is like a rollercoster. However, everytime I try and solve for the parametric equations, I get gravity working "upside down" and my "particle" slows down as it approaches the origin (the bottom of my "rollercoster", whereas it should be speeding up.) Can anyone help? Thanks!

[swansont]

You are dealing with forces, which are vectors. If up is +, down will be -, i.e. the gravitational acceleration is -9.8 m/s^2

[elfmotat]

You can parametrize the curve as simply [math]\mathbf{r}(t)=(t,t^2,0)[/math]. The tangent vector to the curve is then [math]\mathbf{r}'(t)=(1,2t,0)[/math], and the acceleration vector is [math]\mathbf{r}''(t)=(0,2,0)[/math]. The tangent component of the acceleration to the curve is the dot product of the acceleration vector with the unit tangent, the normal component is therefore:

 

[math]a_\perp = \frac{|\mathbf{r}'\times \mathbf{r}''|}{|\mathbf{r}'|}[/math]

 

So we have:

 

[math]a_\perp (t) = \frac{|(1,2t,0)\times (0,2,0)|}{|(1,2t,0)|}=\frac{|(0,0,2)|}{\sqrt{1+4t^2}}=\frac{1}{\sqrt{t^2+1/4}}[/math]

 

Plotting a graph:

 

 

This looks as we'd expect. The maximum acceleration occurs at t=0, when the point is at the minimum of the y=x² graph, and it tapers off in either direction.

Edited March 29, 2013 by elfmotat

[Caesius]

That actually helps a lot. Thanks!