Ok so your understanding is absolutely correct...while people have given quite a few good answers...i would like to present you my point of view...

acceleration=dv/dt         
now dv/dt can be represented as   (dv/dx)*(dx/dt)                
dx/dt=V                        
Thereforce                
a=vdv/dx               
adx=vdv              
of intergarting with resepect to displacement we get           
v²-u²=2as  where s is the displacement
Multiply both sides by mass 'm'     
mv²-mu²=2mas         
1/2mv²-1/2mu²=mas        
mas is now defined as work 'W'
and 1/2mv²-1/2mu² is defined as change in kinetic energy KE
This is the derviation of the famous work energy theorem.....I hope this helps you