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

In summary: 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.
  • #1
Brilli
48
0

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=vy/vx.
I don't know how to proceed ahead.
 
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  • #2
Brilli said:
I don't know how to proceed ahead.
Make a sketch.
[edit] Don't you think ##\alpha = \theta ## ?
Brilli said:
initial moment angle of velocity =90 degrees
What does this mean ?
 
  • #3
@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.)
 
  • #4
Brilli said:

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=vy/vx.
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?
 

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

1. What is path tracing and why is it important?

Path tracing is a rendering technique used in computer graphics to simulate the behavior of light in a virtual scene. This is important because it allows for the creation of realistic and accurate images by tracing the path of light from its source to the camera.

2. How does path tracing solve for time and motion with inclined planes?

Path tracing solves for time and motion with inclined planes by simulating the interactions between light and objects in a scene, taking into account the geometry and material properties of inclined planes. This allows for accurate rendering of shadows and reflections on inclined surfaces, resulting in more realistic and believable images.

3. What are some challenges with implementing efficient path tracing?

Some challenges with efficient path tracing include finding the balance between accuracy and speed, as well as dealing with complex scenes and lighting situations. Additionally, optimizing the algorithm to run efficiently on different hardware and platforms can also be a challenge.

4. How does efficient path tracing compare to other rendering techniques?

Efficient path tracing is considered to be one of the most physically accurate rendering techniques, as it takes into account the behavior of light in a scene. It also produces high-quality images with realistic lighting and shadows, but it can be computationally expensive compared to other techniques such as rasterization.

5. What are some potential applications of efficient path tracing?

Efficient path tracing has many applications in the field of computer graphics, including film and animation production, video game development, and architectural visualization. It can also be used in scientific and engineering simulations for accurate visualization of physical phenomena and processes.

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