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.

\begin{equation} 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 \end{equation} 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

  • $\begingroup$ 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)) $ $\endgroup$
    – Jatin
    Oct 17 '20 at 10:36
  • $\begingroup$ @Jatin What would be the standard equation? $\endgroup$
    – hwood87
    Oct 17 '20 at 11:30
  • $\begingroup$ $v_f^2= v_i^2-2g(f(x_f)-f(x_i))$ .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. $\endgroup$
    – Jatin
    Oct 17 '20 at 11:32

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