I'd be grateful for some help with this problem I am trying to solve.
Let's say that I have an object travelling at a velocity v. I want that object to come to a halt in time t AND travel exactly distance d within that time.
So if we are at t0 when we are at velocity v and apply the brakes, the distance travelled since I applied the brakes should be d and the time taken to cover d should be t and the velocity at that point, 0.
How should I decelerate?
The specific details of my problem are that I have an object travelling at 1498 (let's say m/s) with a distance left to go of 601 (let's say metres) and 2.535 seconds left.
If I concentrate on v apply a constant deceleration, then
a = -v/t
and the distance I would travel would be
d = vt/2 = 1898.7, much higher than the 601 I have.
It seems to me that I need some kind of sigmoid-like curve.
Edit: based on others' questions / comments, perhaps some background might help. I am an artist who is trying to make a kinetic sculpture that mimics the motion of breathing. According to research I have come across the motion of inhaling is similar to a sigmoid curve, whilst exhaling is like an exponential decay. However, a true exponential decay requires massive acceleration, so I thought of synthesising this by modelling the initial phase with the maximum acceleration that the mechanics of my system will allow. I then chose an arbitrary point that I would refine by experimentation and the decelerate from there to reach the end point (lungs empty) within time t, before then beginning to 'inhale'. I thought that this pragmatic approach might be simpler. Ha!
The numbers I have quoted are based on my simulations in Excel, but I've changed the units to SI for simplicity of explanation.