# Derivative of Position Vector at Specified Time

## Homework Statement

My homework problem is a proof in orbital mechanics, but I'm not looking for specific help on that just yet, I'd like to work through it on my own. In doing so however, I'm having a hard time conceptualizing the idea of derivatives of vectors at a specified time. If r is a general position vector, and r0 is the position vector at time t0, and the same applies for velocity vectors v and v0, it seems to me that the derivatives of each of the vectors specified at time 0 should be 0, because these values are constant. But I also don't see how that can be true because v0 should be the derivative of r0.

## Homework Equations

r=ar0+bv0 where a and b are scalar functions of time.

## The Attempt at a Solution

If I attempt to take the derivative of the above equation, I'm not sure whether or not I can say the derivatives of r0 and v0 are 0, leaving me with v=a'r0+b'v0 or not.

Thanks for any help guys, sorry if this is a bit of a dumb question but it's really messing with my head right now. Cheers

Vectors are analogous to their scalar cousins. Just because an initial position is fixed doesn't imply the initial velocity must be zero, or that the initial acceleration must be zero. Consider throwing a ball from a rooftop. The initial position is fixed but not zero. The initial velocity is not zero because it's thrown, and the initial acceleration is not zero because gravity doesn't go away 