Robert M.

asked • 04/26/19

distance as a function of initial velocity and constant power

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.

1 Expert Answer

By:

Michael D. answered • 06/28/19

Tutor
New to Wyzant

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