Conceptual difficulty with friction and inertia Consider this -
A box lies on the floor of a bus which starts accelerating. Two cases arise
$1.$ If the static friction is great enough, the box accelerates with the bus with acceleration equal to that of the bus in the direction of force
$2.$ If the static friction is not great enough, the box tries to maintain its inertia and then moves with the bus after kinetic friction kicks in.
My question is if my reasoning in the second case is not flawed, the value of kinetic friction is greater than that of static friction (this follows since there are no other forces acting on the block except friction), which is wrong.
Where is my reasoning flawed?
P.S. - In the second case, my reasoning is the static friction is in $0$, so no forces acts on the object initially, but since the floor moves away, kinetic friction increases, which causes the object to move with the bus.
 A: Let's consider both cases in the frame of the road. Let the bus have acceleration $a$.

*

*The box sticks to the bottom of the bus. Hence, the box accelerates with with the bus, and its acceleration is also $a$. This acceleration is due to the static friction force.

*The box slips and doesn't quite accelerate as quickly as the bus. Its acceleration is less than $a$. This acceleration is due to the kinetic friction force.

Since the box experiences a smaller acceleration in scenario (2), it experiences a smaller force. Hence, the kinetic friction force is less than the static friction force.
A: You are wrong in thinking that kinetic friction is greater than static friction. When you start pushing against an object, at first it it does not move, As you increase the force, eventually it starts to move. Once it's moving, you can decrease the force, while still keeping the box moving.
This explains the use of Anti-lock Braking Systems (ABS) in cars. When a wheel is not skidding, its point of contact with the ground is static. When you brake with ABS, sensors detect when a wheel stops turning, indicating skidding. The system then automatically releases the brakes. This allows the wheel to stop skidding and re-establish static friction. Doing this enables the car to sustain maximum braking power.
You can test this if you have a car without ABS. On a wet or icy road (without obstacles, preferably), apply more and more force on the brake pedal. When the wheels lock up it will feel as it the car suddenly shoots forward. Release the brake and try again more gently for more braking force. Before ABS was introduced, this was routine practice for experienced drivers.
A: 

*

*If the static friction is great enough, the box accelerates with the bus with acceleration equal to that of the bus


This is correct, and the sentence should end here. The box and the bus have equal acceleration, and acceleration includes direction. Adding the words “in the direction of force” here does not make sense.



*If the static friction is not great enough, the box tries to maintain its inertia and then moves with the bus after kinetic friction kicks in.


This seems to suggest that the box first remains stationary, but after some time, kinetic friction passes some threshold and the box starts moving. Everything about this description is incorrect.
If the static friction is not great enough, the box slips. When the box slips, there is no static friction at all, only kinetic friction. This kinetic friction is proportional to the normal force (in this case, constant). In any case, the kinetic friction does not depend on how much the box has slipped. [1] The box does not accelerate as much as the bus, but it does accelerate, and this acceleration begins immediately.
[1] Unless this slippage causes the box to move to a section of floor with a different coefficient of friction.
A: The flaw in your reasoning is that you have confused two different situations in case 2.
You have described one situation in which the bus has constant acceleration, and the force of friction is not enough to match the acceleration of the bus. In this case, DanDan0101's answer is completely correct: the box will slide backward forever.
But the other situation is one where the box starts sliding, then eventually stops. This result is entirely possible - you've probably seen it happen in real life, which is why you're asking about it. So what's different about this situation that makes the box eventually stop?
The answer is that the acceleration of the bus isn't constant.
The bus accelerates for a certain amount of time, and during this time the box slides because the friction force can't accelerate the box enough to keep up. Then the bus stops accelerating and moves at a constant velocity. Now the box's acceleration due to friction is greater than the bus's (because the bus's acceleration is 0), and the box will stop sliding once it has accelerated to match the bus's velocity.
