Joys of Time Remapping - How to auto ramp layer speed.

I have a ton of video clips that start as still images, play through, then
freeze on the last frame.

I would like to use time remapping to ease into motion. For example based on the
layer's in point, start from a still then ramp up to full speed over .5 second.
Play the clip at 100% speed, then ramp down to a freeze frame where the out
point is.