1. **State the problem:** We are given the formula $$\frac{v^2}{rg} = \tan \beta$$ and need to understand or manipulate it. 2. **Explain the formula:** This formula relates velocity $v$, radius $r$, gravitational acceleration $g$, and angle $\beta$ through the tangent function. It is often used in physics for circular motion on a banked curve. 3. **Rearrange the formula to solve for $v$:** $$\frac{v^2}{rg} = \tan \beta$$ Multiply both sides by $rg$: $$v^2 = rg \tan \beta$$ 4. **Take the square root of both sides to solve for $v$:** $$v = \sqrt{rg \tan \beta}$$ 5. **Explain the result:** Velocity $v$ depends on the radius $r$, gravitational acceleration $g$, and the angle $\beta$. The tangent of $\beta$ shows how the banking angle affects the velocity. This formula is useful for calculating the speed at which a vehicle can safely travel on a banked curve without relying on friction.