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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.

Qmechanic
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Subbi reddy dwarampudi
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  • When you accelerate the bodies you are putting the energy into them. And when you stop them, you're taking the same amount of energy back out. So how is energy not conserved? – lemon Mar 30 '16 at 09:36
  • so you are telling the amount of energy i get from both the bodies E1 and E2 will be equal even there is a large time difference...? – Subbi reddy dwarampudi Mar 30 '16 at 09:41
  • after t sec the velocity is v, after 10t the velocity will be 10v....Now if i stop the first body at t1 and absorbed energy E1 from it and second body at t2 and absorbed energy E2....how E1 and E2 will be equal? because one is travelling with " v" and other is travelling with "10v" – Subbi reddy dwarampudi Mar 30 '16 at 09:44
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    No... But if you accelerate b2 for a longer period then you are putting more energy into it...! – lemon Mar 30 '16 at 09:46

1 Answers1

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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

Gremlin
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