Momentum and Collisions - Mission MC10 Detailed Help

A small fish with a mass of 'M' is at rest. A big fish with a mass of '3M' is swimming towards it with a speed of 30 cm/s. The big fish opens its mouth and catches the little fish. After lunch, the two fish move together with a speed of ___ cm/s.
(Note: Your mass and speed 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, the small fish is at rest and the total momentum of the system in present in the big fish. 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 fish). So in effect, there was only one object moving before the collision and only one object moving after the collision (the two fish are being considered as a single object). The difference is that the mass of the moving object has increased - from 3M to 4M. That is, before the collision, just a fish with mass 3M was moving; after the collision an object (the two fish combined) with mass 4M was moving. The amount of mass which is moving has increased by a factor of 4/3. For total system momentum to be conserved, the velocity of the object must decrease by a factor of 4/3. So the post-collision velocity is the original velocity divided by 4/3.
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.