# Momentum conservation question (answer provided)

#### physics120

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

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

3. The attempt at a solution

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

Homework Helper
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 dont understand where they got "m1v2 +m2v2 = (m1 + m2) * v2" ESPECIALLY THE "m1v2" part. How did they obtain that?

I dont 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

Homework Helper
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, im understanding what they are trying to do now.
Thank-you for your help.

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