Solving for Velocity in a Momentum and Kinetic Energy Problem

[flyingpig]

Breaking rules of Algebra?

Homework Statement

I am copying this problem from my textbook. It is about momentum and Kinetic energy

By rearranging the top equation, we have PART A
v₁'=v₁‐kv₂'

v₁'²+kv₂'²=v₁²

Now we can substitute the top equation into the bottom: PART B

(v₁‐kv₂')²+kv₂'²=v₁'²
v₁²‐2kv₁v₂'+k²v₂'²+kv₂'²=v₁'²

After the cancellations, we are left with PART C

‐2v₁+kv₂'+v₂'=0

The Attempt at a Solution

My problem lies in part B. According to Part A, v₁'=v₁‐kv₂'
But in part B, (v₁‐kv₂')²+kv₂'²=v₁'², see the problem? v₁‐kv₂' has been plugged into v₁'² and it reappeared again on the right hand side.

Right I need help. KEEP IN MIND THAT "K" IS THE RATIO OF M₂/M₁
ALSO PLEASE SHOW ME HOW TO GET TO PART C

 

[Pengwuino]

What do you mean it reappeared again on the right hand side? It looks fine. I'm not sure about the jump to part C though...

 

[flyingpig]

What do you mean it reappeared again on the right hand side? It looks fine. I'm not sure about the jump to part C though...

Shouldn't it have been this?

(v₁‐kv₂')²+kv₂'²=v₁²

 

[Pengwuino]

Yes, you're right, I need glasses. It has to be unprimed on the right side. Sounds like a typo (prime's are easy to mess up ). The jump to part C makes sense now.

 

[flyingpig]

Yes, you're right, I need glasses. It has to be unprimed on the right side. Sounds like a typo (prime's are easy to mess up ). The jump to part C makes sense now.

What should I do? What the general solution?

I am suppose to get a result of this

v₁'=(m₁‐m₂)v₁/(m₁+m₂)
v₂'=2m₁v₁/(m₁+m₂)

 

[Pengwuino]

Where are these masses coming from? You have part C, it should be immediate what v2' is and from the first equation, you can determine v1'.

 

[flyingpig]

Where are these masses coming from? You have part C, it should be immediate what v2' is and from the first equation, you can determine v1'.

It's a momentum question. Basically it states an object with velocity collides a stationary object that is heavier than the object clashing in. The result is both objects repelling and heading the opposite direction of each other. I am suppose to find the velocity using the conversation of energy - KE.

 

[Pengwuino]

I mean what is the equation for the masses? I assume it's k = something about masses.

 

[flyingpig]

I mean what is the equation for the masses? I assume it's k = something about masses.

Sorry, I don't know what you are talking about? K (in post 1) is the ratio of M2/M1

 

[Pengwuino]

So k=m2/m1. Part C shows you exactly what v2' is so you should be able to solve for it immediately with no tricks or substitutions other then k=m2/m1. To solve for v1', use the very first equation you have after you solve v2'.

 

[flyingpig]

So k=m2/m1. Part C shows you exactly what v2' is so you should be able to solve for it immediately with no tricks or substitutions other then k=m2/m1. To solve for v1', use the very first equation you have after you solve v2'.

Can you show me the algebra...? Because I am still stuck on Part B.

 

[Pengwuino]

You were correct in noticing it was a typo in the third and subsequently 4th line. The $$v_1 '^{^2 }$$ should have been a $$v_1 ^2$$ on the right side. The line after it is also incorrect. You have the right idea to plug in v1' from the first equation. Now expand the left side and its a simple cancellation. Subtract $$v_1 ^2$$ from both sides and you get part C and you can solve for v2' immediately.