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# Physics and Collisions Help

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## Introduction to Collisions

When two objects strike each other because they are in relative motion and their paths cross at exactly the right time, a collision is said to occur.

## Conservation Of Momentum

According to the law of conservation of momentum , the total momentum contained in two objects is the same after a collision as before. The characteristics of the collision do not matter as long as it is an ideal system . In an ideal system, there is no friction or other real-world imperfection, and the total system momentum never changes unless a new mass or force is introduced.

The law of conservation of momentum applies not only to systems having two objects or particles but also to systems having any number of objects or particles. However, the law holds only in a closed system , that is, a system in which the total mass remains constant, and no forces are introduced from the outside.

This is a good time to make an important announcement. From now on in this book, if specific units are not given for quantities, assume that the units are intended to be expressed in the International System (SI). Therefore, in the following examples, masses are in kilograms, velocity magnitudes are in meters per second, and momentums are in kilogram-meters per second. Get into the habit of making this assumption, whether the units end up being important in the discussion or not. Of course, if other units are specified, then use those. But beware when making calculations. Units always must agree throughout a calculation, or you run the risk of getting a nonsensical or inaccurate result.

## Sticky Objects

Look at Fig. 8-2. The two objects have masses m 1 and m 2 , and they are moving at speeds v 1 and v 2 , respectively. The velocity vectors v 1 and v 2 are not specifically shown here, but they point in the directions shown by the arrows. At A in this illustration, the two objects are on a collision course. The momentum of the object with mass m 1 is equal to p 1 = m 1 v 1 ; the momentum of the object with mass m 2 is equal to p 2 = m 2 v 2 .

Fig. 8-2 . ( a ) Two sticky objects, both with constant but different velocities, approach each other, ( b ) The objects after the collision.

At B , the objects have just hit each other and stuck together. After the collision, the composite object cruises along at a new velocity v that is different from either of the initial velocities. The new momentum, call it p , is equal to the sum of the original momentums. Therefore:

p = m 1 v 1 + m 2 v 2

The final velocity v can be determined by noting that the final mass is m 1 + m 2 . Therefore:

p = ( m 1 + m 2 ) v

v = p /( m 1 + m 2 )

## Bouncy Objects

Now examine Fig. 8-3. The two objects have masses m 1 and m 2 , and they are moving at speeds v 1 and v 2 , respectively. The velocity vectors v 1 and v 2 are not specifically shown here, but they point in the directions shown by the arrows. At A in this illustration, the two objects are on a collision course. The momentum of the object with mass m 1 is equal to p 1 = m 1 v 1 ; the momentum of the object with mass m 2 is equal to p 2 = m 2 v 2 . Thus far the situation is just the same as that in Fig. 8-2. But here the objects are made of different stuff. They bounce off of each other when they collide.

Fig. 8-3 . ( a ) Two bouncy objects, both with constant but different velocities, approach each other, ( b ) The objects after the collision.

At B , the objects have just hit each other and bounced. Of course, their masses have not changed, but their velocities have, so their individual momentums have changed. However, the total momentum of the system has not changed, according to the law of conservation of momentum. Suppose that the new velocity of m 1 is v 1a and that the new velocity of m 2 is v 2a . The new momentums of the objects are therefore

p 1a = m 1 v 1a

p 2a = m 2 v 2a

According to the law of conservation of momentum,

p 1 + p 2 = p 1a + p 2a

and therefore:

m 1 v 1 + m 2 v 2 = m 1 v 1a + m 2 v 2a

The examples shown in Figs. 8-2 and 8-3 represent idealized situations. In the real world, there would be complications that we are ignoring here for the sake of demonstrating basic principles. For example, you might already be wondering whether or not the collisions shown in these drawings would impart spin to the composite mass (in Fig. 8-2) or to either or both masses (in Fig. 8-3). In the real world, this would usually happen, and it would make our calculations vastly more complicated. In these idealized examples, we assume that no spin is produced by the collisions.

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