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"A crate is sitting in the center of the back of a flatbed truck. As the truck accelerates to the east, the crate moves with it, not sliding on the bed of the truck. In what direction is the friction force exerted the truck on the crate?"

The answer is "east". My question is, if the friction force didn't exist, which way would the crate move towards? Would it go west with respect to the flatbed and fall off? If so, is that the reason why the friction force has to go east? Because it opposes the relative motion? What if the crate actually moved east instead of falling off? Is that possible? Would the necessary friction force in that case go west? And ultimately, what determines the direction the crate would move towards? Since the acceleration is eastwards, wouldn't the crate naturally tend to move eastwards, and then the friction force should have to go west?

EDIT: I think when I first asked this I was getting lost by confusing the scenario presented in the question with other possible scenarios. In the question, what happens is the crate moves with the truck, which, if I was in the truck next to the crate, would seem like the crate didn't move at all, as if it was always at rest, right? Even though it is moving. But to be clearer, the question now is: What would need to happen in a situation where the truck accelerates East and the crate slides off? What would the forces acting on the crate be? And what would happen in a scenario where the truck accelerates East but the crate doesn't "move with it" but rather moves like this: enter image description here

Number one represent the position it's in before the truck accelerates East and number two represents where it would be after the truck has accelerated. To the person in the truck next to the crate, it will appear to have moved. This is the situation I was imagining when I asked "What if the crate actually moved east instead of falling off? Is that possible? Would the necessary friction force in that case go west?" In fact, this is what I was originally imagining when reading the question. And wondering what forces would be needed to act on the crate in this case.

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    $\begingroup$ The questions says the crate moves east .... therefore there must be force in that direction! If there was no friction the box would not move ... relative to the road .... and the truck would be well on its way east. $\endgroup$
    – PhysicsDave
    Jun 19, 2019 at 2:14

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Friction opposes relative motion between the two surfaces in question. As the truck accelerates in the eastward direction, the crate, without friction, would tend to stay at rest with respect to the ground i.e. move west with respect to the truck, like you said. Therefore friction acts to oppose this sliding by acting to the east.

As pointed out in the comments, you also know the friction force on the crate is to the east because this is the only horizontal force responsible for causing the crate to move with the truck. By Newton's second law it must be that friction acts to the East on the crate. In order for the crate to move westward relative to the Earth it would need a net westward horizontal force applied to it. The truck accelerating under the crate cannot supply this.

All of your other questions are hard to follow due to changing scenarios. It looks like you are getting yourself confused with all of the different scenarios in your head. If this doesn't address your main issue I would suggest editing your question to be clearer.

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    $\begingroup$ It seems like I am getting confused with all the different scenarios, and also with frames of reference. But I think I'm understanding this situation better now: in order for the crate to move WITH the truck, it must have the same acceleration as the truck, and since the friction force is the only force acting on the crate, it must have the same direction as the acceleration. Therefore the friction force acts to the East. $\endgroup$
    – Agus
    Jun 19, 2019 at 3:49
  • $\begingroup$ I'm now wondering, what would happen in a scenario where the crate slides off? For someone who is observing the truck from outside, it would seem like the crate moves West while the truck moves East. Does that mean that there is a force that pushes the crate West? And where could that force come from? You mentioned that if there was no friction, it would move West with respect to the truck, which, as I understand, is the same as saying that it would slide off. But if there is no force acting on the crate, how can it slide, how can it move at all? $\endgroup$
    – Agus
    Jun 19, 2019 at 3:52
  • $\begingroup$ @Agus If the crate slides off an observer on Earth would not see it move west. If there was some actual sliding they would see the crate move East (just not moving as far as the truck), and in the extreme frictionless case no movement at all. In the frictionless case the truck just moves out from under the crate and then the crate falls to the ground. $\endgroup$
    – BioPhysicist
    Jun 19, 2019 at 4:23
  • $\begingroup$ I see, that makes things clearer, thanks. $\endgroup$
    – Agus
    Jun 19, 2019 at 5:12
  • $\begingroup$ @Agus Please up vote any answers that are helpful, and please accept an answer that sufficiently answers your question. $\endgroup$
    – BioPhysicist
    Jun 19, 2019 at 10:16
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well this one is not that tough as it seems first thing is, if you are taking truck as your frame of reference, then you are in an non-inertial frame of reference

now let's sit on the truck and observe the crate.As truck is an non-inertial Frame , i must consider a pseudo force on crate(mass of crate times acceleration of frame of reference i.e.the truck) , in direction opposite to acceleration of frame of reference(to west as Frame of Reference is accelerating to east)

considering first the case of friction ,it must oppose the relative motion, therefore oppose the pseudo force i.e. friction must be toward east. the net external force on crate is zero . it doesn't moves at all.

considering the case without friction, there is no other force then pseudo force, therefore crate accelerate toward west and fell of the truck

The scenario you are thinking can never happen as the there is no net force in east direction on crate

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