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## Homework Statement

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

## 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: [itex] \kappa = \frac{1}{R} [/itex]. Using the definition of an ellipse: [itex] \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 [/itex], I solved for 'y' and ended up with: [itex] y = b \sqrt{1- \frac{x^2}{a^2}}[/itex]. Then, I used the definition of curvature [itex] \kappa = \frac{\frac{d^2y}{dx^2}}{(1+(\frac{dy}{dx})^2)^\frac{3}{2}} [/itex]. 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 [itex] \kappa = \frac{1}{R} [/itex], therefore [itex] R = \frac{1}{\kappa} [/itex]. So I substituted my equation for R (1 over kappa) into the equation for centripetal acceleration( [itex] a = \frac{v^2}{R} [/itex] and used the given [itex] v_0 = 360 \frac{km}{hr} [/itex]. 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...