In this set of exercises the student builds a

computational model of a hanging mass$-$spring system that is constrained to move

in $1$D$,$ using the simple Euler and the Euler$-$Cromer numerical schemes. The student

is guided to discover, by using the model to produce graphs of the position, velocity,

and energy of the mass as a function of time, that the Euler algorithm does not

conserve energy, and that for this simple oscillatory system, a modified algorithm

$($Euler$-$Cromer$)$ is necessary to avoid artificial behavior in the model