Sliding sideways along a plane

[cs2018]

Homework Statement

[/B]
This problem is from David Morin's Introduction to Classical Mechanics, Chapter 3, Problem 7:
A block is placed on a plane inclined at angle θ. The coefficient of friction between
the block and the plane is μ = tan θ. The block is given a kick so that it initially
moves with speed V horizontally along the plane (that is, in the direction perpen-
dicular to the direction pointing straight down the plane). What is the speed of the
block after a very long time?

Homework Equations

I've set up the equations and shown that the force (along the plane) from the effect of gravity is equal to the friction force (along the plane). The issue is understanding what it is saying about the initial motion.

The Attempt at a Solution

What does it mean to move "horizontally along the plane"? Is this the same as moving along the plane? That's usually how these questions are set up. One complication is that the title of the question is "Sliding sideways along a plane" and this is a different title from question 6, which is "Sliding down a plane" and can easily be solved.

The part in parentheses (that is, in the direction . . .) suggests that the block is moving perpendicular to the (inclined) plane because the motion is perpendicular to the direction pointing straight down the plane. Or is this perpendicular to the level ground because the direction pointing straight down the plane is the vertical direction?

The solution is not helpful and says at the end that after a long time, the block is essentially moving down the plane. How else is it supposed to move if it doesn't move down the plane? Shouldn't the block always be moving down the plane if the acceleration down the plane is zero, as the solution states?

 

[kuruman]

Hi cs2018 and welcome to PF.

This is how I interpret the picture.
Imagine the x-axis along the incline, positive is downhill. Imagine the z-axis perpendicular to the incline pointing away from the surface. The block has an initil velocity that is along the y-axis which is, of course, perpendicular to both the x and z axes. Gravity is perpendicular to the y-axis only; it forms angle θ with respect to the negative x-axis and angle 90^o-θ w.r.t. the negative z-axis.

 

[Orodruin]

A plane has two directions. One horizontal and one is the slope direction. Note that friction will not cancel gravity exactly from the beginning.

 

[cs2018]

Hi kuruman and Orodruin,

Thank you very much for the help. It seems obvious, but just didn't get it. The comment about friction not cancelling gravity right away was very helpful for getting the solution. Thank you again. :)