How Do You Solve for tanB in an Elastic Collision Problem?

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Homework Help Overview

The discussion revolves around an elastic collision problem involving two spheres, where the original poster seeks to derive an expression for the tangent of the angle of reflection (tanB) in terms of the masses and the angle of incidence (A). The context includes the conservation of momentum and energy equations relevant to the collision.

Discussion Character

  • Exploratory, Assumption checking, Mathematical reasoning

Approaches and Questions Raised

  • The original poster attempts to resolve momentum equations and has derived an expression for tanB but is uncertain about the next steps. They question the utility of the energy equation and express difficulty in simplifying their equations.
  • Some participants suggest considering the energy equation and equating angles A and B, while others inquire about the validity of methods when A does not equal B.

Discussion Status

Participants are actively engaging with the problem, offering hints and exploring various approaches. There is no explicit consensus on the best method, and multiple interpretations of the equations and their implications are being discussed.

Contextual Notes

The original poster notes that they are at a specific part of the problem and are constrained by the requirement to express tanB in terms of mass and angle A. There is an indication of complexity due to multiple velocity terms and potential assumptions about the relationship between angles A and B.

Welshy
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Homework Statement


A sphere of mass m1 traveling at velocity u1 collides elastically with a stationary sphere of m2. The particles move away with v1 and v2 respectively.

I'm at part 4, of 5. It asks us to resolve the conservation of momentum equations normal and tangential to the post-collision velocity of m2. I obtained these equations using the diagram that's provided. The part I'm stuck on is where they ask us to obtain an expression for tanB involving only the masses and angle A.

A = angle of incidence
B = angle of reflection

The 5th part asks "Hence when does the angle of incidence equal angle of reflection?". So I know the resultant equation should be something along the lines of:

tanB = ((m1 + m2)/m2)*tanA

as m1 should be << m2

Homework Equations


m1u1cosA = -m1v1cosB + m2v2

m1u1sinA = -m1v1sinB

m1u12 = m1v12 + m2v22

The Attempt at a Solution



I found terms for v1 and equated them eventually getting:

tanB = ((m1u1cosA)/(m1u1cosA - m2v2))*tanA

But now I'm stuck. Any pointers?
 
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Welcome to PF!

Welshy said:
m1u1cosA = -m1v1cosB + m2v2

m1u1sinA = -m1v1sinB

Hi Welshy! Welcome to PF! :smile:

What about the energy equation?
 


tiny-tim said:
Hi Welshy! Welcome to PF! :smile:

What about the energy equation?
I used that in an earlier part of the question and when I tried to use it here it only complicated things by giving me more velocity terms which I couldn't cancel.

Any pointers on how it could be used - or even if my existing methods are on the right track?
 
Hint:

i] put A = B

ii] use (u1 + v1) and (u1 - v1), and the energy equation. :smile:
 
tiny-tim said:
Hint:

i] put A = B

Sorry, I don't think I worded the question right. The bit I'm stuck on needs B = [mass terms]*A so I can't do that yet. But I'll try the other hint. Thanks.
 
tiny-tim said:
Hint:

i] put A = B

ii] use (u1 + v1) and (u1 - v1), and the energy equation. :smile:

I'm having no joy with this at all. Each of u1 and v1 can have 2 different equations. That gives a total of 8 combinations for (u1 - v1)(u1 + v1) and I'm sure I've tried them all.

I don't think I can make them cancel that way when A doesn't = B . :/
 
Any more hints that anyone can give? Is tiny-tim's method valid for A not equal to B? And, if so, where will I pull my speed equations from?

The sheer algebra of this just has me stumped. :/
 

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