Momentum and Collisions - Mission MC10 Detailed Help

A roller skater with a mass of '2M' is moving along the ice with a speed of 6.0 m/s. She collides head-on with a stationary skater who has a mass of 'M.' After the collision, the two skaters grab onto each other (without spinning or losing their balance) and move with a speed of ____ m/s. Enter your answer accurate to the second decimal place.
 
(Note: Your mass values are selected at random and are likely different from the numbers listed here.)


 
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In this question, like most questions in this mission, the collision is a perfectly inelastic collision with both objects moving together as a single unit after the collision. Before the collision, one skater is at rest and the total momentum of the system in present in the other skater. After the collision, the two objects move together and in this sense, the total momentum of the system is present in a single object (the combination of the two skaters). So in effect, there was only one object moving before the collision and only one object moving after the collision (the two skaters are being considered as a single object). The difference is that the mass of the moving object has increased - from 2M to 3M. That is, before the collision, a skater with mass 2M was moving; after the collision an object (both skaters together) with mass 3M was moving. The amount of mass which is moving has increased by a factor of 3/2. For total system momentum to be conserved, the velocity of the object must decrease by a factor of 3/2. So the post-collision velocity is the original velocity divided by 3/2.
 
An alternative method of analysis involves writing an equation in which you express the total momentum of both objects before the collision as being equal to the total momentum of both objects after the collision. See the Know the Law section. Since the mass is not known, the momentum will have to be expressed in terms of the mass M. Ultimately the variable M will cancel from both sides of the equation and the velocity can be calculated as a numerical quantity.


 
The Law of Momentum Conservation:
If a collision occurs between objects 1 and 2 in an isolated system, then the momentum changes of the two objects are equal in magnitude and opposite in direction. That is,
 
m1 • ∆v1 = - m2 • ∆v2

The total system momentum before the collision (p1 + p2) is the same as the total system momentum after the collision (p1' + p2'). That is,     
 
p1 + p2 = p1' + p2'

Total system momentum is conserved in an isolated system.