Question

Derive s= ut+1/2 at2 and v2-u2=2as

Answer 1

acceleration is taken as constant
we know a = dv/dt
                 dv = a dt   
  Integrating both sides with proper limits 

\int\limits^u_v \, dv= \int\limits^0_t {a} \, dtv∫u​dv=t∫0​adt 

 

[v]^v_u=a[t]^t_0[v]uv​=a[t]0t​ 

v-u=atv−u=at 

v=u+atv=u+at 

second equation

a= dv/dt x dx/dx
a = v dv/dx
v dv = a dx 

  Integrating both sides with proper limits 

\int\limits^v_u {v} \, dv = \int\limits^s_0 {a} \, dxu∫v​vdv=0∫s​adx 

\int\limits^v_u {v} \, dv =u∫v​vdv= 


[\frac{v^2}{2}]^v_u =a[x]^s_0[2v2​]uv​=a[x]0s​ 

v^{2}-u^{2} = 2asv2−u2=2as