In an isolated system, two roller skaters, each with a mass of 100 kg, collide. Skater 1 was initially at rest, while Skater 2 was moving at 6 m/s. They move off together. What is their speed?

Question

jamesj · Answer

You need to to use conservation of momentum.

jamesj · Answer

Let p1 be the momentum of the first skater before the collision on p1' afterwards.  Likewise p2 and p2' for skater 2.

Then by conservation of momentum
$$ p_1 + p_2 = p_1' + p_2' $$

Now, use that the definition of momentum to find the joint velocity afterwards.

anonymous · Answer

so 6m/s?

jamesj · Answer

what's the definition of momentum?

anonymous · Answer

mass x velocity

jamesj · Answer

Yes.  So p1 = ? and p2 = ?

anonymous · Answer

0 and 6ms?

jamesj · Answer

$$ p_1 = 0 \ kg.m/s $$ yes.  

But 
$$ p_2 = m_2.v_2 = ... what? $$

anonymous · Answer

10m/s?

jamesj · Answer

what's m2 = ?
what's v2 = ?

anonymous · Answer

3 m/s?

jamesj · Answer

$$ v_2 = 6 m/s $$
$$ m_2 = 100 kg $$
Hence
$$ p_2 = m_2.v_2 = (100 kg)(6 m/s) = 6000 kg.m/s $$

Therefore the total momentum before the collision is
$$ p_1 + p_2 = 0 + 6000 = 6000 $$

Now what about afterwards?  What do you know about v1' and v2' ?

anonymous · Answer

is it 6 m/s...that's all i need  to know..

jamesj · Answer

No.  That's wrong.  After the collision, they both move off together so their velocities are the same.  Call it v:
$$ v = v_1' = v_2' $$

Hence the total momentum after the collision is
$$ p_1' + p_2' = m_1v_1' + m_2v_2' = (m_1 + m_2)v $$

anonymous · Answer

0!

jamesj · Answer

Now, set that equal to the total momentum before the collision and you can solve for v.

anonymous · Answer

0 m/s!

jamesj · Answer

No.  That can't possibly be right.  If they are both stationary, v = 0, then their total momentum is zero.  But we know it must be equal to 6000.

anonymous · Answer

so 10??

jamesj · Answer

Work the algebra here.  Set that last expression for the total momentum equal to the total momentum before the collision.

anonymous · Answer

it has to be one of them, its a multiple choice question.

anonymous · Answer

can you please just tell me the answer i'll ask my teacher tomorrow

jamesj · Answer

You're just guessing the answers.  I'm trying to show you how to find it.

By Conservation of Momentum, momentum before and after the collision are equal:
$$ p_1 + p_2 = p_1' + p_2' $$
We know that
$$ p_1 + p_2 = 6,000 \ kg.m/s $$
We also know that
$$ p_1' + p_2' = (m_1 + m_2)v $$
where v is their shared velocity after the collision.  Hence we have that
$$  (m_1 + m_2)v = 6000 $$

Now solve for v.

jamesj · Answer

**correction: p1 + p2 = 600, because m2.v2 = (100)(6) = 600.  Hence you have that

$$ (m_1 + m_2)v = 600 $$

anonymous · Answer

10 and 6

jamesj · Answer

Now, what's the value of
$$ m_1 + m_2 $$?

anonymous · Answer

well one is 6...so the other has to be 10!

jamesj · Answer

they both weight 100 kg.  Hence
$$ m_1 + m_2 = ... what? $$
and hence what is the solution to
$$ (m_1 + m_2)v = 600 \ kg.m/s $$

anonymous · Answer

if they weigh 100, it must be 6

jamesj · Answer

$$ m_1 = m_2 = 100 \ kg $$
hence
$$ m_1 + m_2 = 200 \ kg $$
therefore
$$ (200 \ kg) v = 600 \ kg.m/s $$
hence
$$ v = ...what? $$

anonymous · Answer

3!!!!!!!

jamesj · Answer

v = 3 m/s, yes.  Makes intuitive sense as well, as that's half the original velocity.

anonymous · Answer

god i'm stupid. thank you!

jamesj · Answer

Now, do yourself a favor, and write this solution out again on a blank piece of paper.  When you can do that without looking at this or anything else, then you understand the problem.

anonymous · Answer

just did. thank you!