Momentum and Collisions - Mission MC10 Detailed Help

Two carts are at rest upon a track and connected by a compressed spring. The spring is released, pushing upon the two carts such that it propels them in opposite directions. One cart has a mass of '2M' and is propelled with a speed of 30 cm/s. The second cart has a mass of 'M' and is propelled 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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This question pertains to an explosion between two carts of different mass. Before the explosion, both carts are at rest. So each cart has 0 units of momentum and the total amount of momentum possessed by the system of two carts before the explosion is 0 units. If total system momentum is conserved, then the total momentum of the two carts is 0 units after the explosion. But how can this be if both carts possess momentum? How can each individual cart have momentum yet the sum of their momentum values be 0 units? Quite easily! If momentum is a vector and the sum of the vectors is 0, then each object must have an equal amount of momentum and in opposite directions. So     
m1• v1 = - m2 • v2

The product of their mass and velocity must be equal in magnitude. So whichever cart has the smallest mass must be moving with a greater velocity in order to have the same momentum. In fact, the cart which has one-half the mass must have two-times the velocity. The ratio of the cart's masses is inversely related to the ratio of the cart's velocities.
An alternative method of analysis involves writing an equation in which you express the total momentum of both objects before the explosion as being equal to the total momentum of both objects after the explosion. 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 or explosion 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 or explosion (p1 + p2) is the same as the total system momentum after the collision or explosion (p1' + p2'). That is,     
p1 + p2 = p1' + p2'

Total system momentum is conserved in an isolated system.