Show that the time it takes to go from \(x_{1}\) to \(x_{2}\) (in one direction) is equal to $$ t_{2}-t_{1}=\int_{x_{1}}^{x_{2}} \frac{d x}{\sqrt{2(E-U(x))}} $$

Question: Show that the time it takes to move from position \(x_1\) to position \(x_2\) in one direction is given by the integral \(t_{2}-t_{1}=\int_{x_{1}}^{x_{2}} \frac{d x}{\sqrt{2(E-U(x))}}\), where \(E\) is the mechanical energy and \(U(x)\) is the potential energy. _ans_ To show this, we used the conservation of mechanical energy principle to derive an expression for the velocity in terms of the potential energy \(U(x)\) and total mechanical energy \(E\). We then replaced velocity with its definition as the time derivative of position and isolated the time differential. Finally, we integrated both sides of the resulting equation to arrive at the expression for the time it takes to move from position \(x_1\) to position \(x_2\).