# Equation for the velocity of a roller coaster at every point

I have been issued a task to create a roller coaster comprised of a piecewise function. In my research I have come across the an equation to calculate the final velocity of the cart found at https://www.teachengineering.org/activities/view/ind-1996-frictional-roller-coaster-design-project-calculus.

$$$$v_f^2= v_i^2-2g\big(f(x_f)-f(x_i)\big)-\frac{4}{7}g \mid f(x_f)-f(x_i)\mid$$$$ My questions: Is anyone familiar with this equation or the associated differential equation? How is it derived? How do I find a friction-less case? Any help would be appreciated

• This equation is not a standarad equation and gives a reasonable estimate. Derivation is here.An equation without friction would be $v_f^2= v_i^2-2g(f(x_f)-f(x_i))$ Oct 17 '20 at 10:36
• @Jatin What would be the standard equation? Oct 17 '20 at 11:30
• $v_f^2= v_i^2-2g(f(x_f)-f(x_i))$ .This one is a standard kinematic equation(without friction). You might know it as $v^2 = u^2+2as$ where $a$ is the acceleration and $s$ is the displacment. Oct 17 '20 at 11:32