So, in (x,y) coordinates, a vector V with components (Vx,Vy) could be written
V = Vxxu + Vy yu|
The Decelerating Bus
Let us examine the forces on some passenger inside the bus; the x direction is down, y is in the direction in which the bus is moving. Two forces are involved: the weight F1=mg xu, pulling the passengers down, and the reaction F2 of the seat, which does not allow any motion in that direction. As long as the bus moves in a straight line and with a constant speed, these two are the only ones that matter and we get as condition of equilibrium
F1+ F2 = 0
What happens when the brakes are applied? Let us look first from the frame of reference of the outside world. If the bus accelerates forward, the acceleration is
a = a yu
When the brake is applied, therefore
a = − a yu
and the forces on a mass inside must obey Newton's law
F1+ F2 = − ma yu
If equilibrium is to be maintained, the forces must change. The weight F1 is fixed, but to balance horizontal forces, the reaction force F2 must change. For instance, the passenger may grab the seat in front, pushing her or his body back, with a force equal to what the acceleration added on the right side of the equation.
To see how the situation looks in the frame of the bus, we add +ma yu to both sides. On the right now, what is added equals what is subtracted, leaving zero, so the equation becomes
F1+ F2 + ma yu = 0
This may be interpreted in the frame of the decelerating bus as follows. For forces to stay in equilibrium, all forces (as before) must add up to zero, but now they must include an inertial force mayu pushing forward, in the direction of yu. Such an inertial force is only felt in the moving frame. You may call it a "fictitious force" if you wish. But when you need a seatbelt or an airbag to stop it from throwing you through a windshield, it seems real enough!
If you ever take a trip on a jetliner, notice how during take-off you are pushed back in your seat. That is the inertial force acting in the frame of the accelerating airplane. In a decelerating bus, you are pushed forward, but in an accelerating airplane you are pushed back. For a simple experiment, take with you a weight on a string (e.g. a fishing sinker or a stack of screw-nuts tied to the end of the string) and let it hang before take-off, defining the "down" direction. During take-off, the string will slant backwards, by perhaps 5-10 degrees.
All this is rather mild compared to the inertial forces felt by astronauts on a rocket or on the space shuttle during launch. The rocket accelerates at about 2-3g, so the added force felt by the astronauts, in their frame of reference, is 2-3 times their weight.