When the bus accelerates from rest: When a bus abruptly starts moving, the passengers tend to slide backward due to inertia, as the passenger's inertia tends to oppose the bus's forward motion. Hence, the passenger tends to fall back when the bus accelerates forward.
This is because of inertia of motion. The part of our body, which is in contact with the bus, changes the direction of motion. However, the rest of the body tries to move in the same direction as before because of inertia of motion.
One of the most important symmetries in all of physics is known as time-reversal symmetry. Put simply, it says that the laws of physics obey the same rules whether you run the clock forward or backward.
Mainly in order to maintain schedule. Traffic on the streets is highly unpredictable. There is a certain degree of uncertainty associated with travel times from one stop to another even when the bus way is grade separated from other traffic. In mixed traffic, there is no way to predict travel times at all.