# The Three $$e$$'s

Geometry Level 5

Consider an ellipse on the Cartesian Plane. Its x-intercepts are $$\sqrt{2}$$ and $$-\sqrt{2}$$, and its $$e=\frac{1}{\sqrt{2}}$$. Now, it intersects with $$y=e^x$$ at two points. Let these points be $$(x_1,y_1),(x_2,y_2)$$, with $$x_1<x_2$$. Now, consider the ellipse to be solid, and let acceleration due to gravity act in the direction of the negative y-axis. Let a particle be projected from $$(x_1,y_1)$$ to $$(x_2,y_2)$$. The particle collides with the ellipse with an $$e=1$$. The particle bounces off the ellipse, and lands back onto the ellipse at $$(x_3,y_3)$$. Find $$x_3$$ to $$4$$ decimal places.

In this question, the first $$e$$ refers to the eccentricity of the ellipse. The second $$e$$ refers to the Euler's Constant$$\approx2.7\ldots$$. The third $$e$$ refers to the coefficient of restitution of collision.

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