Same energies are transferred to 2 objects. But they gain different amounts of energy. Where is the mistake?

Question

Situation #1.

In space, an object is moving with a constant velocity in an inertial reference frame. The object is connected to a winch (winch “A”) with a rope. Winch “A” is in front of the object, and it is resting in the inertial reference frame. The rope’s end that is on the object’s side is not simply tied to it. It is connected to another winch (winch “B”) on the object. Winch “B” winds the rope in a way that it keeps the rope streached, but it doesn’t draw it (ie. the winch’s action doesn’t affect the object’s velocity). Then, winch “A” starts winding the rope’s other end. After working for some time it stops. As it worked, it consumed some amount of energy. That energy is transfered to the object.

Situation #2.

Evereything is as in the situation #1 except these two:

1. The object is moving faster;
2. Winch “B” winds the rope correspondingly faster to keep it streached.

In both the situation the energies that are used to power winches “A” are same and they are transferred to the objects (ie. the objects gained same energies).

But in the situation #2 the object travells longer distance in the period the winch “A” works (because it was moving faster) . Hence the work done by the force that draws the object to winch “A” is greater. That means, in situation 2, the object gains more energy than in situation 1.

Answer 1

In both the situation the energies that are used to power winches “A” are same and they are transferred to the objects (ie. the objects gained same energies).

The mistake is neglecting the work done by winch B. In scenario 2 the work done by winch B is greater than in scenario 1. The sum of the energy used by both winches is the increase in KE.

The problem you are running into is due to a misunderstanding expressed here:

Winch “B” winds the rope in a way that it keeps the rope streached, but it doesn’t draw it (ie. the winch’s action doesn’t affect the object’s velocity).

You seem to believe that this implies that winch B does no work, which is not correct in general. The tension at winch A is equal (assuming a massless cable) to the tension at winch B. So the only way for B to reel in cable and yet do no work is if the tension is 0, in which case the object gains no energy and no work is done by winch A either.

Let's check this mathematically. Suppose that winch A reels in the cable at a speed of $v_A$ with a tension force of $F_T$. Suppose that the object is of mass $m_O$ and moving at speed $v_O$. In order to maintain the cable stretched winch B must reel in the cable at a speed of $v_B=v_O-v_A$. Now, the power provided by winch A is $P_A=F_T\cdot v_A$ and the power delivered to the object is $P_O=F_T\cdot v_O$. In general $P_O\ne P_A$. However, the power provided by winch B is $P_B=F_T\cdot v_B$ so together we have:

$P_A + P_B = F_T\cdot v_A + F_T\cdot v_B$

$P_A + P_B = F_T\cdot (v_A + v_B)$

$P_A + P_B = F_T\cdot (v_A+ v_O-v_A)$

$P_A + P_B = F_T\cdot v_O $

$P_A + P_B = P_O$

Therefore winch B does work, and the power from both winches together is equal to the power delivered to the object, and so energy is conserved.