Dynamics S3 slides 5-6: Roller coaster and shuttlecock at the bottom of the path (ENGR5004)

Dynamics S3 slides 5-6: Roller coaster and shuttlecock at the bottom of the pathFrom lectureCurrent spec2:25 marks5 min

ENGR5004 Dynamics Session 3, slides 5-6 (Applications of a particle in curvilinear motion (2) and (3))

A roller coaster travels down a hill whose path can be approximated by a function $y = f(x)$ . It starts from rest and increases its speed at a constant rate. Similarly (slide 6), a shuttlecock launched by a robot's firing system follows a path approximated by $y = f(x)$ .

Formulas you may need

$\rho = \dfrac{[1 + (dy/dx)^2]^{3/2}}{|d^2y/dx^2|}$ (learn this)
$a_t = \dot v$ , $a_n = \dfrac{v^2}{\rho}$ , $a = \sqrt{a_t^2 + a_n^2}$ (learn this)
Constant $a_t$ : $v = v_0 + a_t t$ , $v^2 = v_0^2 + 2a_t(s - s_0)$ (learn this)

(a)
How can we determine the velocity and acceleration at the bottom of the path?
[4]
(b)
Why would we want to know these values?
[1]