newtonian mechanics - Law of conservation of energy and time

Consider the following in a space (with out any external influences like gravitation)

Let say i have two identical bodies b1,b2 and i applied a force F on them to make them accelerate a. After t time they achieved a velocity of v.

Now lets say i absorbed all the energy present in b1 which is E1 at time t.

And i absorbed all the energy in b2 which is E2 at time 2t.

Now i believe E2>E1 as b2 will have more velocity than b1.

How the Law of conservation of energy is working in here.

Answer

When you accelerate by applying a force, you are doing work on the bodies. The work done will be equal to the energy gained by the bodies. The expression for this is $\Delta E = W = \vec{F} \cdot \vec{d}$, where $\vec{d}$ is the distance moved, and I'm assuming a constant force magnitude and direction.

The distance travelled under constant acceleration, starting from rest, is $d = \frac{1}{2}a t^2$, and the work done is therefore

$W = F\,d = m\,a \, \frac{1}{2}a t^2 = \frac{1}{2}m a^2 t^2$.

However, remember that $v = a t$, so we can re-write this as

$\frac{1}{2} m v^2$,

which is just the expression for the kinetic energy in the body.

So, in your question, for each body, you absorb the same amount of energy stopping them as you used to put them in motion. This is true for each body independently, and therefore for the system as a whole