Question

derive the third equation of motion : v2 - u2 =2as

Answer 1

From 1st equation of motion,
                                             
                                             V = U + AT
                                            Therfore, V - U = AT 

We know that 
   
                     S = (U + V / 2 )×T
    
                   ∴ S/T = U + V / 2
                    ∴U + V = 2×S/T
                    ∴ U + V = 2S / T
So Substituting eq1 and eq2, 
       
                                 (V - U ) (V + U ) = AT ×2S/T
  
                                  ∴V² - U² = 2AS.

Answer 2

We know,

1st equation of motion,

v = u + at

⇒ v - u = at ---------- equation 1

average velocity  = s / t

average velocity = u + v / 2

u + v / 2 = s / t

s = ($\frac{V+U}{2}$ ) × t

s / t = (u+v) / 2

⇒ u + v = 2s / t  ------------------- equation 2

multiplying equation 1 and equation 2

⇒ (v-u) (v+u) = (at)(2s/t)

⇒ v²- u²= 2as

So, third equation of motion is v²- u²= 2as.