SAT Physics Momentum and Impulse - Conservation Of Momentum

SAT Physics Momentum and Impulse - Conservation Of Momentum

CONSERVATION OF MOMENTUM
When objects interact in a closed system the total momentum of the objects is conserved. This means that the total momentum at the beginning of a problem must equal the total momentum at the end of a problem. This is known as the conservation of momentum. It is most often used to solve problems involving collisions and explosions. Mathematically, con­servation of momentum can be expressed as:
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ENERGY IN COLLISIONS
Kinetic energy shares the same variables as momentum. Changes in momentum can cause changes in kinetic energy. When it comes to conservation of energy, only total energy is conserved. Kinetic energy is only one of many energies comprising total energy, and kinetic energy can change during a problem. Therefore, kinetic energy is not always conserved.

Kinetic Energy Lost
During a collision, the kinetic energy after the collision will either be equal to or less than the kinetic energy before the collision. When kinetic energy decreases during a collision, kinetic energy is said to be lost. Conservation of energy dictates that energy cannot be lost, which means that the lost kinetic energy must have gone somewhere. During a collision, objects contact each other, causing the molecules in the objects to vibrate. The microscopic, and invisible, random motion of molecules is known as thermal energy. If you touch a nail after striking it repeatedly with a hammer, you will feel the increase in temperature. The thermal energy generated during the collision then radiates into the environment as heat. Kinetic energy lost in collisions becomes equal to the heat generated. The equation for kinetic energy during a collision is very similar to the equation for conservation of energy. The main differ­ence is the subtraction of kinetic energy lost, Klost, from the left side of the equation.
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Collisions are categorized into two main categories, elastic and inelastic, depending on whether or not kinetic energy is lost during the collision.

Elastic Collisions
In elastic collisions, objects bounce off of each other. In the process, kinetic energy is con­served. In order for this to occur, the collision must not create any vibrations in the colliding objects, which would imply that the objects never touch each other. Examples may include particles, such as two colliding protons, whose repulsion would prevent them from hitting one another. A larger example would be two objects with a spring mounted on one of them. During the collision, the spring would temporarily store and then release the kinetic energy that would have been lost during the collision. Any energy loss at the spring may be negligible and can be ignored. Very hard objects such as billiard balls or steel spheres will also be nearly elastic with minimal energy loss. For testing purposes, if a problem states that a collision is elastic, then kinetic energy is conserved and there is no kinetic energy lost. Linear momentum is always conserved in any type of collision where NO external forces, such as friction, act.
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Inelastic Collisions
If kinetic energy is lost during a collision, then the collision is inelastic. There are two types of inelastic collisions. In the first, the objects may bounce off of each other. In the second, the objects may stick together. When objects stick together, the collision is said to be perfectly (completely, totally) inelastic. The majority of collisions involve an energy loss, making them
inelastic collisions.

INELASTIC (ORDINARY) COLLISIONS
In ordinary inelastic collisions, the objects bounce off of one another as they do in elastic col­lisions. Momentum is conserved as before, but now kinetic energy is lost.
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Both elastic and ordinary inelastic collisions involve objects that bounce off of each other.How are these two collisions distinguished on an exam? Conservation of linear momentum is the same for both of these collisions. The only aspect that differs is kinetic energy. When the collision is elastic (kinetic energy conserved), exam questions will specifically state that the collision is elastic or will indicate that kinetic energy loss is negligible. When the collision is inelastic or perfectly inelastic, exam questions may focus on solving the amount of kinetic energy that is lost.

PERFECTLY INELASTIC COLLISIONS
In these collisions, the objects fuse to become one larger combined mass. To combine, they touch and vibrate, resulting in lost kinetic energy. The conservation of momentum and kinetic energy lost equations are simplified slightly to account for the combined final mass with a single velocity.
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Inelastic Collisions
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A 10,000-kilogram railroad freight car is moving at 3.0 meters per second when it strikes and couples with a 5,000-kilogram freight car that is initially stationary. What is the resulting speed of the railroad freight cars after the collision?
 
WHAT'S THE TRICK?

In collisions, momentum is conserved. Since the freight cars combine, this is an inelastic collision.
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Explosions
In an explosion, a large mass separates into smaller masses. Aside from events caused by actual explosive devices, there are other examples of explosions: ice skaters pushing each other apart and a rocket launch where the rocket is pushed forward as exhaust gases are pushed backward. Essentially, explosions are the reverse of inelastic collisions.
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Since the initial mass is often stationary, vi = 0, this formula can often be simplified for most exam problems.
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In addition, the masses move in opposite directions. Therefore, one of the two velocities must be negative, resulting in a negative momentum. When this momentum is moved to the other side of the equation, it becomes positive
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Energy is not lost in an explosion. In order for an explosion to occur, energy stored in the sys­tem must be released. In mechanics, this is often accomplished by using a compressed spring between two masses or by having a person throw an object. Positioning a spring between masses and then releasing it converts the potential energy stored in the spring into the total kinetic energy of the now-moving masses.
ΔUs = k1 + k2

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