Question

if ti=0 and tf=t, The final angular speed given by this equation
ωf =ωi+αt .........................1
where ωi is the angular speed of the rigid object at time t = 0.and the angular displacement is:
θf=θi+ ωi t+1/2 αt2……….(3)
where θi is the angular position of the rigid object at time t = 0
from Equations (1) and (3), Prove the following:

ωf2= ωi2+2α(θf -θi)
θf=θi+1/2 (ωi+ωf )t

Answer 1

1)

$$θ_f=θ_i+ ω_i t+\frac{1}{2} αt^2\\θ_f=θ_i+ ω_i t+\frac{1}{2}\frac{ω_f -ω_i }{t} t^2\\θ_f=θ_i +\frac{ω_f +ω_i }{2} t$$

2)

$$θ_f=θ_i +\frac{ω_f +ω_i }{2} t\\2(θ_f-θ_i)=(ω_f +ω_i )t=(ω_f +ω_i )\frac{ω_f -ω_i }{\alpha} \\2\alpha(θ_f-θ_i)=(ω_f ^2-ω_i^2 )$$