Momentum and Collisions - Mission MC6 Detailed Help

In a physics lab, two carts (labeled A and B) are at rest on a low-friction track. A spring-like plunger connects them. The springs are compressed and then suddenly released, exerting explosion-like forces on each of the two carts. Cart A is three times (or one-third) as massive as Cart B. During this 'explosion', the force is ____; the acceleration is ____; the impulse is ____; and the momentum change is ____. Enter the letters of the four answers in their respective order.

Newton's Third Law:
In any interaction between two objects, the objects mutually exert forces upon each other. These forces are equal in magnitude and opposite in direction. So if object 1 exerts a force of F2 on object 2, then object 2 exerts an equal force (in the opposite direction) of F1 on object 1. In equation form
F1 = - F2
The logic begins with Newton's third law (see the Know the Law section). In the explosive interaction between Cart A and Cart B, the forces experienced by the carts are equal in magnitude. That is,   
FA = - FB

Common sense would tell us that these forces endure for the same amount of time. After all, if the forces result from the interaction between the carts, Cart A cannot be interacting with Cart B for a different amount of time than the Cart B is interacting with Cart A. So 
 tA = tB

Mathematical logic applied to the two equations above would lead to the conclusion that the product of F and t for Cart A is equal in magnitude (and opposite in direction) to the product of F and t for Cart B. That is,    
FA • tA = - FB• tB

The above statement means that each cart encounters the same impulse directed in opposite directions. Finally, the impulse is equal to momentum change. If applied to this interaction, then one can conclude that Cart A and Cart B also experience the same momentum change. That is
m• ∆vA = - mB • ∆vB

In conclusion: in the explosive interaction between the two carts, the forces exerted on the carts are equal in magnitude, enduring for the same amount of time to produce an equal impulse for each cart and resulting in an equal momentum change for each cart.

In any interaction between two objects, there are some quantities that are always the same for each object (collision force, collision time, impulse and momentum change) and some quantities that often vary (mass, velocity change and acceleration). A careful student of Physics will keep these quantities straight in their mind. Don't be fooled! While the force, impulse and momentum change are the same for each cart, the velocity change and acceleration will be greatest for the least massive cart.

Each cart experiences the same momentum change (see Minds On Time section). That is, the product of mass and velocity change is the same for each:
m• ∆vA = - mB • ∆vB

For the left side of the equation to be equal to the right side, the cart with the smallest mass must have the largest velocity change. The ratio of their mass is inversely proportional to the ratio of their velocity change.


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