Efficient Path Tracing: Solving for Time and Motion with Inclined Planes

[Brilli]

Homework Statement

A small disk A is placed on an inclined plane forming an angle (alpha) with horizontal and is imparted an initial velocity v. Given coefficient of friction =k and ant initial moment angle of velocity =90 degrees. Find out how long it takes to come ro rest and give the equation of motion.

Homework Equations

The Attempt at a Solution

I found out that acceleration in x-axis is kgcosαcos(θ)/
In y-axis is gsinα-kgcosαsinθ
Tanθ=v_y/v_x.
I don't know how to proceed ahead.

 

[BvU]

I don't know how to proceed ahead.

Make a sketch.
[edit] Don't you think $\alpha = \theta$ ?

initial moment angle of velocity =90 degrees

What does this mean ?

 

[Nathanael]

@BvU I believe the initial velocity v is horizontal (90 degrees with the uphill/downhill direction). If you take the initial velocity to be the -x direction and downhill to be +y, then that is the situation his equations seem to describe.If that’s a correct interpretation then the problem at this point boils down to solving a system of 2 differential equations. (Well first we must eliminate θ, or one of the velocity components I suppose, or even both components if we introduce the total speed.)

Maybe there’s a clever way to solve that system of equations but I don’t see a good way. (I don’t think I can help at this point.)

 

[haruspex]

Homework Statement

A small disk A is placed on an inclined plane forming an angle (alpha) with horizontal and is imparted an initial velocity v. Given coefficient of friction =k and ant initial moment angle of velocity =90 degrees. Find out how long it takes to come ro rest and give the equation of motion.

Homework Equations

The Attempt at a Solution

I found out that acceleration in x-axis is kgcosαcos(θ)/
In y-axis is gsinα-kgcosαsinθ
Tanθ=v_y/v_x.
I don't know how to proceed ahead.

Can you write the differential equations for the x and y directions?
Can you see how to get one differential equation that does not involve θ?