0.18 Phy1160: force and motion -- momentum, impulse, and conservation  (Page 9/14)

Except for the conversion of some kinetic energy into other forms, such as sound energy, the two cars coupled together will possess the total amount of momentum that was possessed by the individual cars prior to the collision.

This typically means that one car speeds up and the other car slows down. During the time frame surrounding the instant of the collision, each carexperiences a change in momentum.

A logical proof of the conservation of momentum

We already know that when two objects interact, each object exerts a force on the other object. The two forces are equal in magnitude and opposite indirection.

We can probably agree that when two objects interact, the amount of time that object-A interacts with object-B is the same as the amount of time that object-Binteracts with object-A.

Therefore, if the forces are equal and opposite, and the times are the same, we can write

Fa*t = -Fb*t

where

• Fa and Fb represent forces exerted on objects A and B respectively.
• t represents the time during which the objects interact.

Impulses acting on objects A and B

You should recognize these terms as the impulses acting on objects A and B. Therefore, the impulses acting on the two objects are equal and opposite.

Impulses are equal and object

You learned earlier that the impulse acting on an object is equal to the change in momentum of the object. If the impulses acting on the two objects areequal and opposite, then the change in momentum experienced by the two objects must be equal and opposite.

We can express this in equation form as

ma*(va2 - va1) = -mb*(vb2 - vb1)

where

• ma and mb are the masses of objects A and B respectively
• (va2 - va1) is the change in velocity of object-A
• (vb2 - vb1) is the change in velocity of object-B

The law of conservation of momentum

This equation is a statement of the law of conservation of momentum. The change in momentum experienced by object-A is equal to and opposite of thechange in momentum experienced by object-B.

Stated differently, the momentum lost by object-A is gained by object-B, or vice-versa.

That being the case, the total momentum possessed by the system containing object-A and object-B remainsunchanged by the interaction of the two objects. The total momentum of the system is conserved.

Bowling ball and bowling pins

Bowling is a game where the players roll a heavy ball down a long smooth wooden platform in an attempt to knock down ten heavy wooden objects (bowlingpins) arranged in a triangle at the end of the platform. The bowling pins are shaped somethinglike a flower vase that is larger at the bottom than at the top. Thus, each bowling pin has a low center of gravity.

When the ball strikes the cluster of bowling pins, there are eleven objects involved in a conservation of momentum process. The momentum that is lost by thebowling ball is distributed among the ten bowling pins, but the momentum is probably not distributed evenly among the bowling pins. There is a lot of chaosat the time of impact with the bowling ball colliding with the pins, pins colliding with other pins, etc. Some of the energy is also converted to sound.