Momentum and Its Conservation - Lesson 3

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

The use of

<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

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.

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

As shown in the diagram, the two players have a total momentum after the collision of

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,

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,

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 -

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>.

1. Q about direction affect if p values of individual players were different.

2. Simple Q with different values of pre-collision p; ask for post-collision p in the x- and y-directions

3. Change player A and B directions and ask about the direction of the impulse on A and on B; repeat for delta p.