Angle of a hanging ball in system trying to approach speed of light A common example of acceleration is a ball hanging from the top of the car. The angle this hanging ball makes from zero is dependent on the acceleration of the car.
What happens as we allow the car to attempt to approach the speed of light at constant acceleration?
My expectation is that since the car cannot reach c, it's acceleration must begin to decrease at some point
From this expectation I would further assume that the angle the ball makes to an observer would decrease back to zero.
Is this expectation correct? Where did I go wrong?
 A: You have come to correct conclusion, but the reasoning is not correct.
Indeed, from the point of view of observer who is staying still the acceleration of the car is decreasing to 0 as the speed of car approaches $c$.
But you can't just use non-relativistic approach to calculate the angle of the thread the object hangs on. At least I do not know how to do it correctly.
Better approach would be first to calculate the position of thread in the frame of reference attached to the car. Looks like it's really simple(*): person inside the car "feels" constant acceleration, so the position of the thread will also be constant. Now we only need to change the frame of reference and find out how this picture looks like in frame of reference which is in rest. There is no need to calculate any forces to do that. Imagine that the body and thread are just painted on the wall of the car. $y_1 - y_0$ remains the same, $x_1 - x_0$ gets smaller because of relativistic length contraction, so the angle becomes smaller.
(*) I am not quite sure about this statement. Yes, the person inside the car would feel constant acceleration. But if the gravity field (which from the point of view of observer is now produced by some "earth" which flies around quite fast) - well, I do not know.
