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By my estimation the equation that describes this motion is given by:

$$Pt = \frac{1}{2}m{ \dot x}^2$$

or

$$\dot{x} = \sqrt{\frac{2P}{m}} \sqrt{t}$$

but this implies:

$$\ddot{x} = \sqrt{\frac{2P}{m}} \frac{1}{\sqrt{t}}$$

So, no matter how small we make the power, we appear to get an infinite acceleration at ##t=0##( or in the case of ##P=0, t= 0##) an indeterminate form ##\frac{0}{\infty}##.

Conclusion: Acceleration from rest under constant power is non-physical. However, with any non-zero initial velocity things seem to be ok with the following:

$$\ddot{x} = \frac{P}{m} \frac{1}{\sqrt{ \frac{2P}{m}t + v_o^2}}$$

Anything interesting here, something I'm missing?