At any moment let the speed be v
Therefore the normal component of acceleration is v2R
Tangential Acceleration:
a=v2/R
vdvds=v2R
dvv=dsR
vv0dvv=s0dsR
ln(vv0)=sR
v=v0es/R and x=R[ln(v)−ln(v0)]
Therefore velocity at one revolution:
v1rev=v0e2πR/R=v0e2π
Now using ds/dt=v
dsdt=v
ds=Rdvv
Rdvv=vdt
∫v1revv0Rdvv2=t0dt
R[−1v1rev+1v0]=t
t=R[1v0−1v0e2π]=Rv0(1−e−2π)