A={-1,0}
B={0,t}

p=A+s*(B-A)

res=Solve[Factor[p.p-1]==0, s][[2]]

k=Together[p/.res]

ParametricPlot[k, {t, 0,1}]

k/.{t->1/2}

dk=Together[D[k,t]]

speed=Simplify[Sqrt[dk.dk], t>0]

Plot[speed, {t, 0,1}, AxesOrigin->{0,0}]