# Help with proving this Kepler's laws based theorem

the proposition is that if r(t) has a constant length (//r(t)// is constant), then prove that r(t) is perpendicular to dr/dt at all t.

I was thinking that if r(t) is just a function of the radius and its magnitude //r// is constant, then its basically talking about a circle? and by saying prove that r is perpendicular to dr/dt is basically asking to prove that the radius vector is always orthogonal to the velocity v(t) vector? I need a mathematical proof of this proposition using vectors. Thank, you

What is d/dt(r.r)?

BTW Kepler hasn't got much to do with it.
Any wiggly path on the surface of a sphere has constant r.

David

Last edited:
I'm not sure what your asking. this is a general proof so i dont know what d/dt(r.r) is. all the information i have is posted.

Last edited:
i got an idea... would the //x(t)// be equal to the length of the curve of x(t) and therefore //x(t)//=integral(x(t).dx/dt) and since //x(t)// is constant the indefinite integral can only=c if the x(t).dx/dt=0 and so they are perpendicular. Can someone plz tell me if all this makes sense?

I'm not sure what your asking. this is a general proof so i dont know what d/dt(r.r) is. all the information i have is posted.

r^2 is constant:

0 = d/dt r^2 = 2 r dot dr/dt

0 = d/dt r^2 = 2 r dot dr/dt

oh ok so you're saying that since //r(t)// is constant then //r(t)//^2 is constant which means (r.r) is constant and that derivative which is a dot product is 0. yeah this was easier than i thought my integral solution also works i think