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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