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

[Welshy]

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?

 

[tiny-tim]

Welcome to PF!

m₁u₁cosA = -m₁v₁cosB + m₂v₂

m₁u₁sinA = -m₁v₁sinB

Hi Welshy! Welcome to PF!

What about the energy equation?

 

[Welshy]

Hi Welshy! Welcome to PF!

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?

 

[tiny-tim]

Hint:

i] put A = B

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

 

[Welshy]

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.

 

[Welshy]

Hint:

i] put A = B

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

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 . :/

 

[Welshy]

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. :/