Derivation of Kinematic Equations of Motion

Choose t_i $\equiv$ 0, x_i $\equiv$ 0, v_i $\equiv$ v₀ , and write x_f $\equiv$ x , v_f $\equiv$ v and t_f $\equiv$ t .

• a = $\em$constant $\Rightarrow$ a = $\bar{a}$ . Then Eq.(2.5) $\Rightarrow$ a = ${\frac{v-v_0}{t}}$ or
   

  v = v₀ + at (7)

• a = $\em$constant $\Rightarrow$ v changes uniformly $\Rightarrow$ $\bar{v}$ = ${\frac{1}{2}}$(v₀ + v). From Eq.(2.1) $\bar{v}$ = x/t . Combining: x = $\bar{v}$t = ${\frac{1}{2}}$(v₀ + v)t . Using Eq.(2.7) we get:
   

  $${\textstyle\frac{1}{2}}$$ (8)

• Eq.(2.7) $\Rightarrow$ t = (v - v₀)/a . Substitute into Eq.(2.8) $\Rightarrow$ x = (v + v₀)(v - v₀)/(2a) or,
   

  v ² = v₀² + 2ax (9)

• Note that only two of these equations are independent.