Vectors of Momenta
We live in a world where one's location can be specified by three dimensions: in Cartesian coordinates, x, y, z. Einstein's revision to Newtonian physics added a 4th dimension, time. And now string theorists suggest there may be as many as 7 additional degenerate dimensions necessary to explain String Theory. To explain many of the multi-dimensional properties it is useful to use the mathematical tool of vectors.
vector is formally defined as an element of a space such as that described by Euclid. Vectors are sometimes referred to by numbered coordinates. Vectors are often represented by rays or arrows, drawn to scale. They are typically represented in print by BOLD font and when hand written, a arrow placed above. Vectors have precise rules such as the following:
- A vector has both magnitude as well as direction. Examples of these are displacement, velocity, acceleration, force, momentum, as well as fields. The vector can be moved to another location as long as it maintains its direction and magnitude.
- Two vectors are equal if they have the same direction and magnitude.
- Vectors can be added by moving the second vector, positioning the tail of the second vector to start at the tip of the first arrowhead. The sum is the single equivalent vector which starts at the tail of the first and extends straight away to the second arrowhead. Additional vectors can be added by successively adding an additional vector to the sum of the previous. Addition is commutative so the order of the addition does not change the final sum.
- A quantity without direction is called a scalar. Examples of scalars are common numbers, time, distance, and mass. A vector can be multiplied by a scalar. For example, a vector multiplied by 3 would be a new vector the same direction as the original, but three times the length. Multiplication by a negative vector reverses the original vector's direction. –A is as long as vector A but in the reverse direction. Subtraction can be considered as adding the inverse of the subtracted vector.
- There are three types of multiplication involving vectors. The second kind of multiplication results in a scalar product from the multiplication of two vectors. This is called a dot product. In this case, the common parallel || direction of the two vectors produces the product. Mathematically the effect is represented by the cosine of the angle between the two vectors:
a • b = ab cos Φ
- The third type of vector multiplication results in a vector product from multiplication of two vectors. This is called a cross product. In this case, the direction of the new product is perpendicular ⊥ to BOTH of the multiplied vectors and the magnitude of the new product depends on the degree that the original vectors were also perpendicular. Mathematically the effect is represented by the sine of the angle between the two vectors:
a x b = ab sin Φ
with the product's direction determined by the progression of a conventional screw twisting from a to b
OR using the flat of one's right hand originally in the direction of a with fingers then curled towards b
with the product pointed the direction of the extended thumb.
The angular momentum introduced in Experiment III-2 is an example of a cross product.
The essential goal is to verify that the sum of all the momenta before a collision is identical with the sum after the collision:
Σ m•vbefore = Σ m•vafter
part I: computer simulations
OK. This may be a bit boring because it strips away all the interesting stuff of a real world collision or explosion. But the success of physics comes largely because once an event or phenomena is stripped to its bare essentials, we can analyze and understand it. THEN, once we understand the fundamental truths, we can turn our attention to the much more exciting REAL events. So we start with events which aren't real at all, but only visually presented by computer simulation. This simulation should allow you to quickly investigate a number of virtual two-dimensional collisions to determine how vectors describe collisions. Of course a computer could be programmed to behave far differently that the real world. So any ideas developed from the simulation should be checked to see if they also represent real world events.
Drew Dolgert, student of Michael Fowler at the University of Virginia, has created a Gnu public licensed Java Applet which simulates the elastic collision when a ball collides with a second ball. The applet provides two views of the event, one from the frame of reference for the originally stationary target, and another from the view of a frame traveling with the calculated center of mass. The position of each ball is periodically recorded during each event. Click the Two-dimensional collision, Java Applet which should open a new tab or window. Choose a language as requested to load the applet. Your browser will need to be able to display Java Applets. If you can't access the Java Applet or get it to work correctly, several captured graphics from the Java Applet can be viewed and analyzed instead.
- Measure displacements for each time interval from position trails left on your screen for both balls before and after the collision.
- Calculate speeds and momenta for both balls
- Construct vectors on paper representing the original momenta and that after collision, reproducing the directions as close as possible.
- Add the momenta vectors before the collision and add the momenta vectors after the collision. Compare to see if the sums of the momenta are conserved (with the experimental error of your procedure). Compare measurements made in each frame of reference. Is there any advantage to either frame?
- Change the angle of collision and the relative masses and repeat the analysis of the momenta.
- As you've watched and analyzed these collisions, have you notices any rules that seem to always be true?
part II: table edge collisions
This may be much slower and more time consuming that the computer simulation, but it should allow confirmation that the computer simulation is in fact identical to real world motion. The approach used is to design a reproducible collision event, and then by repeating the identical event in a number of different situations, to tease out in a relatively easy fashion all the measurements needed to analyze the entire event. With a lot more expensive, specialized apparatus the same analysis could be achieved quicker. But the challenge is to figure out how to do this investigation very well with the minimum of specialized equipment and effort.
- Build a little launcher which can reproducibly roll a marble or ball along a table top. This may be as simple as a folded piece of paper propped up on one end so when the ball is rolled down starting from a given spot on the track, it will reliably gain the same speed and direction.
- Locate a second marble or ball on the level table so when it is struck by the first you can determine the directions both move.
- To get some idea on the speeds, launch the first ball or marble so it has identical speed, but arrange it so it rolls off the table nearly perpendicularly to the edge. Observe where it lands and measure how far that is from a vertical just below the edge of the table. The height of the table determines how long the object is in the air. So if it lands further away from the table, then its speed was faster.
- Arrange the collision to occur again, this time with the struck ball or marble so it will promptly roll off the table, again nearly perpendicular to the edge. Observe where it lands and measure the distance away from the table as a measure of its speed just after the collision.
- Finally, repeat the collision again but arranged so the originally moving ball will promptly roll off the table nearly perpendicular to the edge. Observe where it lands and measure the distance away from the table as a measure of its speed just after the collision.
- Construct momentum vectors representing the momentum of each before and after the collision and attempt to determine if the sum of the momenta before equals the sum of the momenta after the collision.
part III: separating components of a vector
Sometimes it is useful to take apart a vector into the two or more perpendicular components. For example a projectile generally has some motion horizontal and parallel to the earth's surface in addition to some vertical rise or fall. Separating its motion into horizontal and vertical components is rather useful because as a first approximation the vertical motion is governed as if the projectile was just a freely falling object while its horizontal motion may be nearly constant if friction is small. Below are five numbered vectors. #1 has already been separated into its vertical and horizontal components, 1v and 1h. Note that these two components can be added back to the original by moving either so its tail starts at the head of the other component. Practice finding the vertical and horizontal components for the other four vectors.
One of the advantages of separating vectors into perpendicular components is that components of several vectors can be added, subtracted, or otherwise transformed algebraically instead of geometrically by constructing scale drawings. The skills used to work with two dimensional vectors can be extended to work with a greater number of dimensions.
Communicating technical information such as observations and findings is a skill used by scientists but useful for most others. If you need course credit, use your observations in your journal to construct a formal report.