Calculating Power with Constant Force: Deriving P=(F^2t)/m from Newton's 2nd Law

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To derive the power equation P=(F^2t)/m for a particle of mass m moving under a constant force F, start with the definition of power as P=F·v. Using Newton's second law, the acceleration a can be expressed as a=F/m, leading to the velocity v as a function of time t, given by v=at=Ft/m. Substituting this expression for v back into the power equation yields P=F·(Ft/m), simplifying to P=(F^2t)/m. The discussion emphasizes the importance of relating velocity to force and mass through kinematic equations to arrive at the desired power formula. This approach clarifies the connection between force, mass, and power in a constant force scenario.
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A particle of mass m moves from rest at t=0 under the influence of a single constant force F. Show that the power delivered by the force at any time t is P=(F^2t)/m.
I tried using the definition of power and the definition of the scalar product, which probably is not the way to go about doing this. I feel like there's an obvious step or equation that I'm missing here.

P=F.v=|F||v|cos0=Fv
mv(dv/dt)=P
 
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using P =Fv is good. Now you need to find v as a function of F, T, and m using Newton's 2nd law (and the kinematic motion equations if it doesn't jump right out at you).
 

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