Centripetal Acceleration of rocket car

[_N3WTON_]

Homework Statement

A rocket car is constrained to move on an elliptical track (semi-major axis $a$ and semi-minor axis $b$). The car is moving at a constant speed $v_0$. Determine the acceleration of the car in $\frac{m}{s^2}$. $a = 4 \hspace{2 mm} km$, $b = 2 \hspace{2 mm} km$, and $v_0 = 360 \frac{km}{hr}$.

Homework Equations

The Attempt at a Solution

My thought for this problem was to try and find the value of 'R' using the definition of curvature: $\kappa = \frac{1}{R}$. Using the definition of an ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, I solved for 'y' and ended up with: $y = b \sqrt{1- \frac{x^2}{a^2}}$. Then, I used the definition of curvature $\kappa = \frac{\frac{d^2y}{dx^2}}{(1+(\frac{dy}{dx})^2)^\frac{3}{2}}$. Then, I took the first and second derivative of my equation for an ellipse (I will not write them here because it gets very messy). Then, I substituted my second derivative into the equation for curvature. Finally, using the fact that $\kappa = \frac{1}{R}$, therefore $R = \frac{1}{\kappa}$. So I substituted my equation for R (1 over kappa) into the equation for centripetal acceleration( $a = \frac{v^2}{R}$ and used the given $v_0 = 360 \frac{km}{hr}$. However, as you can probably guess, this is extremely messy. Furthermore, I do not see how I am to get rid of the x and y in my equations (the answer is meant to be a numeric value). So I am certain I have done something wrong but I'm not sure what...

 

[Simon Bridge]

What doe "R" represent? You have to say... is the idea to use the equation $a=v^2/R$?
Note: the acceleration of the car won't be a constant since it's rate of change in direction varies but it's speed doesn't.

 

[gneill]

You might get neater expressions for the first and second derivatives if you use implicit differentiation.

 

[_N3WTON_]

What doe "R" represent? You have to say... is the idea to use the equation $a=v^2/R$?
Note: the acceleration of the car won't be a constant since it's rate of change in direction varies but it's speed doesn't.

Sorry, I realize I wasn't very clear. The idea was to use the equation $a = \frac{v^2}{R}$.. I attempted to find R using the equation $\kappa = \frac{1}{R}$. So I solved for R using the curvature equation and then plugged this value into the equation for centripetal acceleration but it is just so messy that I think I probably am not on the right track. I hope this clears things up...

 

[_N3WTON_]

You might get neater expressions for the first and second derivatives if you use implicit differentiation.

I hadn't thought about that but I'll certainly give it a shot, thanks :D

 

[_N3WTON_]

Note: the acceleration of the car won't be a constant since it's rate of change in direction varies but it's speed doesn't.

So in that case I am basically at the solution, I just need to clean things up algebraically I think. I have this bad habit where if my solution looks really messy I just assume I've done something wrong..

 

[Simon Bridge]

I can't tell, because you didn't answer the questions.

 

[_N3WTON_]

I can't tell, because you didn't answer the questions.

Which questions are you referring to? R is meant to represent radius of curvature, is there something wrong with how I've approached the problem?