simple yet good ..

well here is the two liner solution ..

as we know friction opposes relative motion , friction will always act opposite to the body's velocity

magnitude of frition = μN(as it is moving)=tanθmgcosθ=mgsinθ

finally as subash and asish have pointed out .. the velocity would be straight down the plane ..

because at each moment

let velocity be oriented at an angle of β to the horizontal clockwise .

ax=-Fcosβ

ay=Fsinβ + mgsinθ

as the object moves its velocity along x decreases and y increases therby increasing β

as tiime passes we see that at sufficient β ay surpasses ax and after a long time ax will almost have reduced to zero
(as β increases at infinite time we would have β =90 and thus the net force if you calculate comes to be zero)

thus the object will have a constant veocity at infinite time .

further observe that -

acceleration along velocity = -F+mgsinθsinβ
acceleration along Vy = -Fsinβ +mgsinθ

so

md(vy + v)/dt=-mgsinθ + mgsinθsinβ -mgsinβsinθ +mgsinθ =0

thus the quantity v + vy always remains constant ..

at t=0 the above quantity is = V₀ + 0
at infinitely long time when v is vy (no vx)

=v + v

so we have

v+v= V₀ + 0

thus v=V₀/2

Problem forgotten??

bhaiyya! can you provide solution or will we have to wait

im not making much progress here

yes subash and asish the direction given by you is correct ..

this is what i know of concept but clueless how to convert it to equations...

--> friction acts in a direction opposite to the resultant velocity..
--> initially friction acts opposite to horizontal direction.
--> as time increases and speed down the incline increases... the frictional force continually changes direction and becomes more inclined towards the downward velocity
--> After infinitely long time the frictional force will be directed up the incline.
--> kaymant sir's hint is i think the best way.. let me try it.. now only understood his hint

yes satan, you are right again.. now the answer to your original question can be easily guessed [1]. but i suggest ppl to first try it themselves before pondering over the answer given by satan

This is the figure for my solution

how can u write tana=V/gtsinθ ?

friction will affect both horizontal and vertical velocities !!

remember the concepts virang ..

friction opposes relative motion

so intially friction will act opposite to the velocity .. and gravity will act down the plane and hence there will be downward motion ..

For motion downwards Forces are gravity and friction

Noraml = mg
Friction is tanθmgcosθ = mgsinθ

Force due to gravity
= mgsinθ

Since friction is equal to gravitational pull There will be no downwards motion.

The Body will only move horizontally on the plane and come to rest after T = V/gtanθ

nothing at all .. didnt i mention its simple ..

well there may be big ways to do it as u mentioned but the method i am talking about doesnt need these ..

hmm....do we take the components of 'g' ? And then friction also will oppose horizontal as well as vertical motion..

remember friction opposes relative motion .. :)

mayb the friction force will also hav to b resolved into 2 components