Momentum conservation question (answer provided)

[physics120]

Homework Statement

I already have the answer, I just need someone to help me understand the answer for the following momentum question:

Two balls are both moving horizontally to the right on a table. Ball 1 catches up with ball 2 and collides with it. The balls stick together and continue on with velocity vf (balls are still moving horizontally and are still on the table). Which of these statements is true?

a) vf is greater than v1.
b) vf = v1.
c) vf is greater than v2 but less than v1.
d) vf = v2.
e) vf is less than v2.
f) Can't tell without knowing the masses.

The ANSWER is c.

Homework Equations

the law of conservation of momentum: vector Pf = vector pi

The Attempt at a Solution

THE CORRECT ANSWER:

Momentum conservation requires (m1 + m2) * vf = m1v1 +m2v2. Because v1>v2, it must be that (m1 +m2) * vf = m1v1 + m2v2 > m1v2 +m2v2 = (m1 + m2) * v2. Thus vf > v2. Similarly, v2 < v1 so (m1 + m2) * vf = m1v1 + m2v2 < m1v1 + m2v1 = (m1 + m2) * v1. Thus vf < v1. The collision causes m1 to slow down and m2 to speed up.

I got this answer from the back of my physics textbook so I am 100% sure it is CORRECT. Right now, I am studying for my midterm on Friday, and I would REALLY appreciate it if someone could explain the answer to me!

Thank-you

 

[tiny-tim]

Hi physics120!

Momentum conservation requires (m1 + m2) * vf = m1v1 +m2v2. Because v1>v2, it must be that (m1 +m2) * vf = m1v1 + m2v2 > m1v2 +m2v2 = (m1 + m2) * v2. Thus vf > v2. Similarly, v2 < v1 so (m1 + m2) * vf = m1v1 + m2v2 < m1v1 + m2v1 = (m1 + m2) * v1. Thus vf < v1. The collision causes m1 to slow down and m2 to speed up.

hmm … works even if v1 < v2 …

personally, I'd write it vf = pv1 + (1-p)v2,

where p = m1/(m1 + m2),

and then obviously vf lies between v1 and v2.

but anyway, what part of the proof in the book is worrying you? ​

 

[physics120]

Well first, this part: "Because v1>v2, it must be that (m1 +m2) * vf = m1v1 + m2v2 > m1v2 +m2v2 = (m1 + m2) * v2. Thus vf > v2."

I don't understand where they got "m1v2 +m2v2 = (m1 + m2) * v2" ESPECIALLY THE "m1v2" part. How did they obtain that?

I don't really know how they used the law of conservation of momentum properly. Could you please show me an even more detailed step-by-step of the proof they did?

 

[tiny-tim]

The first bit, (m1 +m2) * vf = m1v1 + m2v2, is the law of conservation of momentum.

And then they used:

v1 > v2, so m1v1 > m1v2, so m1v1 + m2v2 > m1v2 +m2v2 = (m1 + m2)v2

 

[physics120]

OK, after looking at the answer over, I am understanding what they are trying to do now.
Thank-you for your help.