Why Does Cramster Simplify Velocity Components Incorrectly in Physics Problems?

[Fellowroot]

Homework Statement

This problem is from Modern Physics by Kenneth Krane 2nd ED

The problem is:

An atom of mass m moving in the x direction with speed v collides elastically with an atom of mass 3m at rest. After the collision the first atom moves in the y direction. Find the direction of motion of the second atom and the speeds of both atoms in terms of v after the collision.

Now my question is not how to solve this problem, it is to try and understand a math step involved in this problem from a solution off of cramster.

Homework Equations

The Attempt at a Solution

On cramster they reduce this:

9v$^{2}_{}$cos$^{2}$(x)+9v$^{2}_{}$cos$^{2}$(x)

to this:

9v$^{2}$

But when I did the problem and got down to their step I have this:

9v$^{2}_{x}$cos$^{2}$(x)+9v$^{2}_{y}$cos$^{2}$(x)

Those velocities are different components an x and a y, which they neglected to distinguish on cramster.

I know that:

3cos$^{2}$(x) + 3sin$^{2}$(x) = 3

But I don't see how they came to their conclusion because the x and y components could be different.

Thanks.

 

[vela]

If your equation is correct, you're right that it can't be reduced. Show us how you got your equation.

 

[Fellowroot]

There is about 3 pages of work to get to that equation but I don't think I have made a mistake. So its pretty hard to post it. Also if I go by their solution they actually get the correct answer in the end but at the same time I'm very sure that they are indeed two different components. Thanks though.

 

[vela]

Well, I'll just say your equation looks wrong. You almost always get v_x or v cos θ, not the combination v_xcos θ. Similarly, for the y-component.

 

[rwisz]

I would first like to acknowledge that understanding the math steps involved in a problem is crucial to fully understanding the concepts of modern physics. It is important to note that the solution provided on cramster may not be the only correct solution and it is always beneficial to cross-check with other sources.

In this case, it seems that the solution on cramster may have made a simplification assuming that the velocities in the x and y directions are equal, leading to the reduction of the equation to 9v^2. This is not necessarily true and it is important to consider the components of velocity separately in order to accurately solve the problem.

It is always important to carefully examine and understand each step of a solution in order to fully grasp the concepts being taught. If you are having trouble understanding a particular step, it may be helpful to consult with your instructor or a peer for further clarification. Additionally, exploring different resources and approaches to solving the problem can also aid in understanding the material.