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Power = Force x velocity

At the initial instant, the velocity = 0, so the power is zero, even though there may be force, but zero power is taken from that force. So how can the object even start moving, if the power is zero, obviously the work done is zero, and without any work or power, it cannot accelerate, so it ought to remain stuck at v=0. It seems to say that the object will accelerate and the velocity will increase from zero even though the initial power absorption is zero.

Let us assume force = 1N, mass = 1kg, there is no friction. I know what f=ma and the equations of motion say, but how is it possible for the object to accelerate with zero initial power drawn from the force? No initial work should mean no increase in KE and no increase in velocity.

If this is true for the very first instant, then the same should be true for all following instants, since the velocity and power still remains zero for each successive instant. So why does the object accelerate with zero initial power? Where does the very first increase in energy come from to change the velocity from zero to delta v?

user1648764
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    But if there's a force, there's an acceleration, and if there's an acceleration, velocity has to change!?! The problem is you are trying to think of this in discrete chunks, when the concepts all apply to continuous situations. – JMac Aug 23 '17 at 19:11
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    You're right nothing ever moves. – Señor O Aug 23 '17 at 19:13
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    Possible duplicates: https://physics.stackexchange.com/q/60480/2451 , https://physics.stackexchange.com/q/111251/2451 , https://physics.stackexchange.com/q/172207/2451 and links therein. – Qmechanic Aug 23 '17 at 22:04

2 Answers2

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You appear to be right that the initial power is zero - but you also need "zero energy" to accelerate, since the kinetic energy goes as $E=\frac12 m v^2$.

If we say power = force times velocity, $P = F\cdot v$, and rate of change of kinetic energy is $\frac{d(\frac12 m v^2)}{dt}=mv \frac{dv}{dt}$, then we can say $F\cdot v = m v \frac{dv}{dt}$. Dividing both sides by $v$, we end up with the usual $F = m \frac{dv}{dt} = m\cdot a$.

The only thing that is tricky here is that you will say "you can't divide both sides by $v$, because $v$ is zero!

To which I would answer - to an observer moving in another frame, the velocity would not be zero so I can safely divide... the underlying equations would still hold true.

Floris
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                                                                                       בס''ד
The question is about

Power = Force x velocity

What that power is - being referenced - is the power of the moving object, once it has been accelerated. That is, once the object is moving, it can do work on other objects as it is brought once again to rest.

Since the velocity is zero to start out with, the object being moved is not storing any power to start with, as expected. It has to be accelerated to store power. With 

Force = mass x acceleration = mass * d/dt velocity
Power = Force x Velocity = mass * d/dt velocity * velocity
Energy = Integral Power dt = mass * 1/2 velocity ^2

That is the energy conveyed to the object is the familiar 1/2 mass * velocity squared as expected.

Stephen Elliott
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