Thorton - Marion Problem 2.40

x[t_] := a(2 α t - Sin[α t])
y[t_] := a (1 - Cos[α t])

ParametricPlot[{x[t],y[t]} /. a -> 1/. α -> 2 π, {t, 0, 2}]

Part (a)

v[t_] = {D[x[t],t], D[y[t],t]}

a[t_] ={D[x[t],{t,2}],D[y[t],{t,2}]}

Tangential acceleration along the velocity vector

v[t](v[t].a[t])/(v[t].v[t])

% // Simplify

Normal acceleration perpendicular to velocity vector

a[t] - v[t] (v[t].a[t])/(v[t].v[t])

%//Simplify

%.% // Simplify

Part (b)

Plot[% /. a -> 1 /. α -> 2 π, {t, 0, 2}]