Mass M with initial velocity =Vo represented by initial energy = Eo => Vo=sqrt(2/M*Eo).
Energy (and velocity) increase as constant power applied:
e(t)=Pt + Eo => v(t)=sqrt[ 2/M* ( Pt + Eo )]
distance s(t) as a function of time t = integral of v(t) dt
s(t)=integral { v(t) } dt
s(t)=integral { sqrt[ 2/M* ( Pt+Eo ) ] } dt
the constant comes out => s(t)= sqrt(2/M) * integral { sqrt [ Pt+Eo ] } dt
this is of the form integral { sqrt [ ax+b ] } dx, which my sources integrate as
integral { sqrt[ ax + b ] dx = (2b / 3a + 2x / 3 ) * sqrt ( ax + b )
therefore: s(t)= sqrt (2/M) * integral { sqrt [ Pt+Eo ] } dt
s(t) = sqrt (2/M) * { [2*Eo ]/ [3*P] + [ 2*t/3 ] * sqrt [Pt+Eo ].
If this s(t) is evaluated at t=0, the answer should be 0 because even with an initial velocity or energy, at time zero the Mass has still not moved any where.
However, this equation evaluated at t=0 is sqrt(2/M) * [2*Eo ] / [ 3*P ] * sqrt ( Eo) ;
which is not 0.
What am I doing wrong?
Thank you.