a2[θ0_,  dθ0_,  θf_,  dθf_,  
        tf_] := ((3*(θf - θ0))/
            tf^2) - ((2*dθ0)/tf) - dθf/tf

a2[a, b, c, d, e]

a3[θ0_,  dθ0_,  θf_,  dθf_,  
        tf_] := ((-2*(θf - θ0))/
            tf^3) + ((dθf + dθ0)/tf^2)

a3[a, b, c, d, e]

viaθ[θ0_,  dθ0_,  θf_,  dθf_,  
        tf_,  t_] := θ0 + (dθ0*
            t) + (a2[θ0,  dθ0,  θf,  
              dθf,  tf]*
            t^2) + (a3[θ0,  dθ0,  θf,  
              dθf,  tf]*t^3)

viaθ[θ0,  dθ0,  θf,  dθf,  tf,  
        t] // TraditionalForm

viaθ[0,  0,  45,  0,  3,  1]

fig46a = 
      Plot[viaθ[-45, 0, 30, 0, 1.25, t], {t,  0,  1.25},  
        Frame -> True,  
        FrameLabel -> {"Time (seconds)",  "Position \
(degrees)"}]

temp[time_] = 
      Dt[viaθ[-45,  0,  30,  0,  1.25,  time],  
        time]

temp[a]

Plot[temp[a],  {a,  0,  1.25},  Frame -> True,  
      FrameLabel -> {"Time (seconds)",  "Velocity \
(degrees/second)"}]

tempb[tt_] = Dt[temp[tt],  tt]

\(Plot[tempb[b], \ {b, \ 0, \ 1.25}, \ Frame -> True, \ 
      FrameLabel -> {"Time(seconds)", \ \*""Acceleration (degrees/(second^2))""}]\)

Path with via points....

segA[time_] := 
      viaθ[-45,  0,  30,  20,  1.5,  time]

apos = 
      Plot[segA[time],  {time,  0, 1.5},  Frame -> True,  
        FrameLabel -> {"Time (seconds)",  "Position \
(degrees)"}]

dsegA[time_] = Dt[segA[time],  time]

avel = 
      Plot[dsegA[a],  {a,  0,  1.5},  Frame -> True,  
        FrameLabel -> {"Time (seconds)",  "Velocity \
(degrees/second)"}]

ddsegA[time_] = Dt[dsegA[time],  time]

\(aaccel = 
      Plot[ddsegA[a], \ {a, \ 0, \ 1.5}, \ Frame -> True, \ 
        FrameLabel -> {"Time(seconds)", \ \*""Acceleration(degrees/(second^2))""}]\)

segB[time_] := viaθ[30,  20,  75,  0,  3,  time]

bpos = 
      Plot[segB[(time - 1.5)],  {time,  1.5, 4.5},  
        Frame -> True,  
        FrameLabel -> {"Time (seconds)",  "Position \
(degrees)"}]

dsegB[time_] = Dt[segB[time],  time]

bvel = 
      Plot[dsegB[a - 1.5],  {a,  1.5,  4.5},  Frame -> True,  
        FrameLabel -> {"Time (seconds)",  "Velocity \
(degrees/second)"}]

ddsegB[time_] = Dt[dsegB[time],  time]

\(baccel = 
      Plot[ddsegB[a - 1.5], \ {a, \ 1.5, \ 4.5}, \ Frame -> True, \ 
        FrameLabel -> {"Time(seconds)", \ \*""Acceleration(degrees/(second^2))""}]\)

Show[apos,  bpos]

Show[avel,  bvel]

Show[aaccel,  baccel]