How to derive the Equation

Question

How to derive Vf^2 = Vo^2+2AD from the equation for acceleration and displacement/distance equation.

Answer 1

Take the displacement equation s = 1/2 ( u + v ) t and the acceleration equation a = ( v - u ) / t.

u = initial velocity and v = final velocity s = displacement

Solve for t in each equation and then equate the results.

2 s / ( u + v ) = ( v - u ) / a ==> v^2 - u^2 = 2 a s

I call this the SUVAT Equation

Answer 2

Start with the relations dx/dt=v and dv/dt=a

Take (dx/dt) / (dv/dt) = v/a

This will result in a dx = v dv

Integrate both sides,

a(x-x_0) = 0.5(V^2 - V_0^2)

Moving terms around results in V^2 = V_0^2 + 2a(x-x_0)