Question

if an object follows the law of conservation of energy, does this mean that the object cannot accelerate?

Answer 1

No, the particle can still accelerate.

Let us take a simple example , a free falling object of mass M starting from rest , from a certain height h under the action of gravity.

At , height h , K.E = 0

P.E = Mgh

Total mechanical energy E = K.E + P.E

= Mgh

When the object falls a distance d , and attains a speed V ,

K.E = 0.5 MV²

= 0.5 M ( 2gd)

= Mgd ( since , V² = 0 +2gd )

P.E = Mg( h-d )

Total mechanical energy at the position

E = K.E + P.E

= Mgd + Mg(h-d )

= Mgh

SO, total energy is conserved but have an acceleration of g .