Realizing my ‘logisitic’ curve would probably not be the best idea, why not just set the acceleration to a constant on keypress, and then verlet integrate? The acceleration set could be computed from Newton’s second law, so ‘friction dampening’ would be easy.
double accelerationX = (keyPressedForceX - frictionForceX) / mass;
double accelerationY = (keyPressedForceY - frictionForceY) / mass;
positionX = positionX + (positionX - lastX) + accelerationX * delta * delta;
positionY = positionY + (positionY - lastY) + accelerationY * delta * delta;
Of course there are better integration methods to use if so inclined.
edit: added sample code