Question

why do you fall in the forward direction when a moving bus brakes to a stop and fall backward when it accelerates from rest

Answer 1

In a moving bus, passengers are in motion along with bus. When brakes are applied to stop a moving bus, bus comes in the position of rest. But because of tendency to be in the motion a person falls in forward direction.

Similarly, when a bus is accelerated from rest, the tendency to be in rest, a person in the bus falls backwards

Answer 2

Due to inertia of motion, your body tends to be in motion, while it is at motion.
So when the bus brakes, it brings your body to rest, hence the body tries to resist the stopping, but it can't, as your feet have already come to rest, hence you move forward a bit, and come back.

It's the reverse case during rest to acceleration, and it's called as inertia of rest.
The person tends to be in rest, but when the bus suddenly moves forward, the body resists motion, hence we tend to fall backwards.

Hope it helps....
^_^