How friction acts on a block moving down a slope moving side to side

[QuantumOscillatorIII]

TL;DR Summary: How friction acts on a block moving down a slope moving side to side

I found this problem, and I've also attached its solution. My question is, if the block wants to move down the slope, and also wants to move side to side and follow the movement of the plane - where does friction act? Since the total friction force is max $ \mu * mg * \cos (\theta) $ part of that will act against gravity, the other part against the side-to-side movement of the plane. The solution says that the part that will act against gravity will be equal to the downward gravitational force - but isn't this a wild assumption? Also, what happens if the downward gravitational force is greater than $ \mu * mg * \cos (\theta) $? Does the solution fall apart then?

 

[haruspex]

My question is, if the block wants to move down the slope, and also wants to move side to side and follow the movement of the plane - where does friction act? Since the total friction force is max $\mu * mg * \cos (\theta)$ part of that will act against gravity, the other part against the side-to-side movement of the plane.

Friction acts to oppose relative motion of surfaces in contact.
The solution assumes constant downslope velocity, $w$. Because the block barely has time to acquire any lateral velocity, the relative velocity in that direction is always $v$.
The overall relative velocity is therefore at angle $arctan(w/v)$ to the horizontal, and that is the direction in which friction will act.

The solution says that the part that will act against gravity will be equal to the downward gravitational force - but isn't this a wild assumption?

That comes from the assumption of constant downslope velocity.
Ignore the small lateral velocity the block acquires and consider the block starting from rest. At first, all the relative velocity is lateral, so no friction opposes the descent. The block accelerates down the slope, but as it does so an increasing fraction of the frictional force will point up the slope. So, provided $\mu>\tan(\alpha)$, there is a limiting downslope speed.

 

[QuantumOscillatorIII]

Friction acts to oppose relative motion of surfaces in contact.
The solution assumes constant downslope velocity, $w$. Because the block barely has time to acquire any lateral velocity, the relative velocity in that direction is always $v$.
The overall relative velocity is therefore at angle $arctan(w/v)$ to the horizontal, and that is the direction in which friction will act.

That comes from the assumption of constant downslope velocity.
Ignore the small lateral velocity the block acquires and consider the block starting from rest. At first, all the relative velocity is lateral, so no friction opposes the descent. The block accelerates down the slope, but as it does so an increasing fraction of the frictional force will point up the slope. So, provided $\mu>\tan(\alpha)$, there is a limiting downslope speed.

Got it, thanks!