Momentum and Its Conservation - Lesson 3 - Collisions in Two Dimensions

# 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. 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. 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 Lesson 3. 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 later 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 the collision.  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

Lesson 2 focused on the application of the law of momentum conservation to the analysis of collisions. According to this conservation law, the total system momentum before the collision is equal to the total system momentum after the collision. 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 isolated system.

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 y-dimensions. 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, Player A is moving south with 300 kg•m/s of momentum; Player B 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 Diagram D, 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 momentum and southern momentum of the system was owned 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 hit-and-stick collisions 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 earlier in this chapter that the impulse-momentum change theorem is often 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. When first thinking about this, you might quickly conclude that there was a southward force on Player A. After all, Player B was moving south and bumped into Player A, thus exerting a southward push  upon Player A. This southward force would lead to a southward impulse upon Player A. After the collision, Player A has a velocity that is both south and east. The southward component of velocity for Player A was not present before the collision. So clearly, Player A has an increase in southward momentum, consistent with the southward impulse. This illustrates that the impulse direction on Player A is equal to the momentum change direction of Player A.

What about Player B? We can apply very similar reasoning to show 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 eastward force upon Player B. This eastward force on Player B causes Player B to experience an eastward impulse. After the collision, Player B is moving east in addition to south. And so Player B has gained some eastward momentum that he did not have before the collision. This means that Player B has an eastward component of momentum change, consistent with the eastward impulse that he experiences. And so once more, the impulse direction is equal to the momentum change direction.

But that's not all. There's more to 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 Newton's third law of motion, 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 southward force upon Player A, there must also be a westward force upon Player A. Player A must encounter both a southward impulse (as already discussed) and a westward impulse. This westward 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, thus decreasing the amount of eastward momentum that it originally had.

If Newton's third law predicts a decrease in eastward 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 eastward impulse (already discussed) and a northward impulse. The northward impulse on Player B must cause a northward momentum change. That is, southward moving Player B loses some of its southward momentum.

The impulse momentum change theorem predicts that Player A loses eastward momentum and gains southward momentum.
It also predicts that Player B gains eastward momentum and loses southward momentum.

So far in our discussion in Lesson 3, we have not focused upon velocities. The focus has been on momentucm changes of individual objects and momentum conservation of the system. We will focus on the topic of velocity and velocity changes in 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?

Question 1:
The example discussed on this page involved a collision between two football players. Before the collision, Player A had 400 units of eastward momentum and Player B had 300 units of southward momentum. Suppose that Player A instead had 300 units of eastward momentum before the collision. What would be the effect of this change upon the post-collision direction of the two players? And why?
a. This would have no effect upon the post-collision momentum and direction since momentum is conserved and never changing.
b. The players would still move both east and south after the collision. But since we don't know the masses, it's impossible to say anything more.
c. The players would move both east and south after the collision at a 45° angle to east. In other words, they would move exactly southeast.
d. Nonsense! Life is so uncertain that you can't even predict whether these days, let alone the result of a collision.

Question 2:
Consider a collision between two objects in which they stick together after the collision and move as a single unit. Suppose Object A has 5 units of westward momentum and Object B has 12 units of northward momentum. How will the system of two objects move after the collision?

Question 3:
Let's suppose the football player collision example included Player A moving northward with a momentum of 280 kgm/s and Player B moving westward with a momentum of 210 kg•m/s. Describe the direction of the impulse on Players A and B.

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