distance as a function of initial velocity and constant power

Question

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.

Answer 1

Looks like you have assumed that E(t) = p x t + Eo is only mechanical energy

Usually we write E(t) = K + U where both are dependent on t.

We also want by convention U(s) to become zero as s increases to infinity.

We can set up U(to) to cancel S(to).....This potential energy should be the energy that our object can return to when the power is removed that is moving it. This is not what you have modeled.