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I am thinking about something that is getting me quite confused. To illustrate what is confusing me let's have a cart moving in the horizontal direction on a track without friction, and a motor which adjusts to keep the kart at a constant speed V in the positive direction. Now, it's snowing so the mass of the cart is increasing at a rate dm/dt. The power the motor must do to keep the cart going at a constant speed is P=(dm/dt)V

(I should emphasize the snow is falling with no horizontal velocity relative to the ground, and the motor is keeping the cart at a constant velocity since it is inclined to slow because of conservation of momentum).

^{2}. But, the change in kinetic energy is dT/dt=d((1/2)mV^{2})/dt=(1/2)(dm/dt)V^{2}. So, if we integrate the power and change of kinetic energy over some interval of time, we find that the interval of work by the motor is twice the change in kinetic energy. Where is the other half of the work going? There is no friction in the problem. I understand why mathematically it doesn't work out as usual where these things are equal (because W≠ΔT in this problem because the integral of force doesn't work out that way because the change in mass), but I am struggling to conceptually understand this scenario.(I should emphasize the snow is falling with no horizontal velocity relative to the ground, and the motor is keeping the cart at a constant velocity since it is inclined to slow because of conservation of momentum).

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