Momentum and Its Conservation - Lesson 3

# Momentum Conservation Times Two

This unit of The Physics Classroom Tutorial has focused on the application of two principles to the analysis of collisions. The principles are the impulse-momentum change theorem and the law of momentum conservation. Every analysis that has been performed in the previous two lessons has involved one-dimensional collisions. The colliding objects have been moving along the same line before and after the collision. For instance, the collisions we have been analyzing have involved two objects which were moving east - both before and after the collision. Or one object was moving north and collided directly head on with a second object moving south before the collision; after the collision, the objects continued to move along the north-south axis. These types of collisions were one-dimensional collisions.

In Lesson 3, we will consider the collision of objects moving in two dimensions. A variety of two-dimensional collision types will be discussed. The collision of two objects moving at right angles to one another is one type of collision. This situation is shown in Diagram A below; after the collision, the two objects stick together  and move as a single object. This type of collision will be the focus of the next page of <a href="u4l3b.cfm">Lesson 3</a>. Another type of collision involves the collision of a moving object with a stationary object as shown in Diagram B. Since the line of motion of the moving object is not directly in line with the center of the stationary object, the two objects move in different directions after the collision. This type of collision is sometimes referred to as a glancing collision. Glancing collisions will be discussed <a href="u4l3c.cfm">later</a> in Lesson 3. Another collision type includes two objects moving at acute angles to one another before the collision and bouncing off each other. This type of collision is shown in Diagram C.

The use of types to categorize these collisions is not the point of this discussion. The collision types are presented here in an effort to get you thinking about what makes a collision a two-dimensional collision. As you can see, a two dimensional collision is a collision in which both objects are moving in different directions either before or after the collision (or both before and after).  Regardless of the collision type, one thing that we can be certain of is that the same principles that governed one-dimensional collisions will also govern two-dimensional collisions. As we will soon see, the impulse-momentum change theorem and the law of momentum conservation can be used to analyze these collisions.

The Law of Momentum Conservation
<a href="u4l2a.cfm">Lesson 2</a> focused on the application of the <a href="u4l2b.cfm">law of conservation of momentum</a> to the analysis of collisions. According to this conservation law, the total system momentum before the collision was equal to the total system momentum after the collision. We could say that momentum is a quantity that is conserved.  This conservation occurs as long as there are no external forces that contribute momentum to or remove momentum from the system. These external forces are forces that act between the objects of the system and other objects not included in the so-called system. A collision in which external forces do not contribute to or take away momentum from the system are said to occur in an <a href="u4l2c.cfm">isolated system</a>.

The law of momentum conservation can also be applied to the analysis of two dimensional collisions. But for two dimensional collisions, one must consider momentum conservation to occur in two dimensions - that is, in the proverbial x- and the y-dimension. So the sum of all the x-momentum before the collision is equal to the sum of all the x-momentum after the collision. And similarly, the sum of all the y-momentum before the collision is equal to the sum of all the y-momentum after the collision.

Example 1
To illustrate this conservation of momentum in two dimensions, consider the following example. Two football players collide in mid-air. Before the collision, one player is moving south with 300 kg•m/s of momentum; the second player is moving east with 400 kg•m/s of momentum. Before the collision, there is both x- and y- momentum. After the collision, there should be the same amount of x- and y-momentum, directed in the same direction as before the collision. After the collision (a.k.a., the tackle), the two players travel together in the same direction. The direction of motion is both south and east, but not equally south as east. In fact, the two players travel together in a more eastern direction than a southern direction.

As shown in the diagram, the two players have a total momentum after the collision of ptotal. This ptotal vector has two components or parts. There is an eastern component and a southern component to the ptotal vector. The eastern component has a magnitude of 400 kg•m/s and the southern component has a magntiude of 300 kg•m/s. Together, these two components add up to the ptotal vector. And individually, these two components demonstrate the law of momentum conservation. The eastern component of the post-collision ptotal vector is equal to the amount of eastern momentum of the system before the collision. And the southern component of the post-collision ptotal vector is equal to the amount of southern momentum of the system before the collision. This is the law of momentum conservation ... in two dimensions. Before the collision, this eastern and southern momentum was possessed by two individual objects. Player A had the east momentum and Player B had the south momentum. After the collision, it is shared between the two players witih each player having both east and south momentum. This is a common trait of an inelastic collision in which the two objects stick together and move as a single unit.

Impulse-Momentum Change Theorem
Now we will investigate the application of the impulse-momentum change theorem to the analysis of a two-dimensional collision. You might recall from <a href="u4l1b.cfm">earlier in this chapter</a> that the impulse-momentum change theorem is used to analyze the before- and after-momentum of a single object involved in a collision. Each object in a collision encounters a change in momentum that is caused by and equal to the impulse that the object experiences. That is,

Impulse = Momentum Change

The impulse experienced by an object is equal to the product of the force exerted upon it and the time over which the force acts - F•t. And the momentum change is the product of the obejct's mass and its velocity change - m•∆v. And so according to the theorem,

F•t = m•∆v

Let's look at the previously discussed football player collision in light of the impulse-momentum change theorem. Player A was moving east before the collision and encountered a force from Player B. It is relatively easy to convince someone that the force on Player A was a southern force. After all, Player B was moving south and bumped into Player A, thus pushing Player A in the southern direction. This southern force leads to a southern impulse upon Player A. After the collision, Player A has a velocity that is both south and east. The southern component of velocity was not present before the collision. So clearly, Player A has an increase in southern momentum, consistent with the southern impulse. This illustrates that the impulse direction on Player A is equal to the momentum change direction of Player A.

But what about Player B? We can apply very similar reasoning to shown that the impulse direction on Player B is equal to the momentum change direction of Player B. Player A was moving east and collided with Player B who was moving south. Upon collision, Player A exerts a eastern force upon Player B. This eastern force on Player B causes Player B to experience an eastern impulse. After the collision, Player B is moving east in addition to south. And so Player B has gained some eastern momentum that he did not have before the collision. This means that Player B has an eastward component of momentum change, consistent with the eastern impulse that he experiences. And so once more, the impulse direction is equal to the momentum change direction.

But that's not all. Here is the rest of the story. It centers around a jingle that we are all familiar with - for every action, there is an equal and opposite reaction. According to <a href="../newtlaws/u2l4a.cfm">Newton's third law of motion</a>, if Player A pushes Player B to the east, then Player B must also exert a westward force upon Player A. In addition to the more obvious southern force upon Player A, there must also be a westward force upon Player A. Player A must encounter both a southern impulse (as already discussed) and a western impulse. This western impulse on Player A must cause Player A to experience a westward momentum change. That is, eastward moving Player A must slow down in the eastern direction. Some of the velocity in the east direction must have been lost. So far in our discussion in Lesson 3, we have not focused upon velocities. This will be the topic of the next part of Lesson 3. Exactly how can we use momentum conservation to predict the post-collision velocities of the individual objects within an isolated system?

If Newton's third law predicts a decrease in eastern momentum of Player A, what does if predict of Player B? If Player B pushes Player A to the south, then Player A must also exert a northward force upon Player B. This northward force upon Player B corresponds to a northward impulse. So Player B must experience both an eastern impulse (already discussed) and a northern impulse. The northern impulse on Player B must cause a northernmomentum change. That is, southward moving Player B must somehow lose some of its southern velocity. But without any knowledge of the before- and after- collision velocities of these two players, it is not possible to substantiate this claim. And so once more, we must ask the question how can we use momentum conservation to predict the post-collision velocities of the individual objects within an isolated system? This will be discussed <a href="u4l3b.cfm">next</a>.