r[t_]={2*t+5,-t^2,(1/3)t^3};

Assuming[t>0,Refine[Norm[r[t]]]]

v[t_]=r'[t]

a[t_]=v'[t]

vn[t_]=Assuming[t>0,Refine[Simplify[Norm[v[t]]]]]

at[t_]=vn'[t]

atot=Assuming[t>0,Refine[Simplify[Norm[a[t]]]]]

an[t_]=Simplify[(Sqrt[atot^2-at[t]^2])]

ParametricPlot3D[r[t],{t,0,2},
AxesLabel->{x,y,z},PlotStyle->{Blue,Thick}]