Collisions Of Elastic Spheres

This lab studies the collision between two spheres. The data you obtain from your measurements will be used to verify momentum conservation. This experiment will require efficient organization of a lot of data. You will use the spreadsheet program Excel to help you organize and analyze all your data.

Apparatus

You will use the following equipment to perform this set of experiments:

Inclined ramp to propel incident sphere with a known velocity, supporting device for the target sphere, two steel spheres, two glass spheres, paper and carbon paper for locating position of spheres when they hit the floor, plumb bob to locate position of incident sphere before collision, clamps, balance, support rods, and string.

Objectives

In this experiment we want to study the collision of a moving sphere with a stationary sphere and examine the concepts of conservation of energy and momentum in such a process. You are to do all three parts of the experiment. It may be best to get data for all three parts before going into detailed analysis of one part. You may also want to find a way to divide up the analysis. In either case, come prepared and plan your lab time well.

                          Figure 1

Procedure

Part I: Determination of the Initial Momentum

We will give an initial velocity to a ball by letting it roll down a ramp from a fixed height h. We locate the point below which it leaves the ramp and begins to fall, using a plumb bob and mark it on the floor. We record the point where the ball hits the ground by placing carbon paper on the white paper where the original impact occurs (so that the carbon paper will leave a mark on the white paper).

Obtain five successive impact points by repeatedly releasing the sphere from the same initial position, h.

Using analysis of the motion of the ball after it leaves the ramp, find the relationship between V0, the sphere's horizontal velocity as it leaves the ramp, and D, the point where the object lands (see Figure 1). Using this relationship and the measured distance find V0. Also find the uncertainty in V0. Compare this to what you would predict from conservation of energy of the ball coming down the ramp. Discuss any discrepancies you observe. Measure the mass of the ball. From the measured mass and velocity, determine the ball's linear momentum.

Part II: Collisions between Equal Mass Balls

                                  Figure 2

In this part we will investigate collisions between spheres of equal size and mass. To do so, we will place a target sphere on the support screw at the base of the ramp. The height of this screw should be adjusted so that the incident or projectile sphere will just pass over the screw without touching it when no target sphere is supported on it. Why? The position of this supporting screw can be moved sideways to vary the impact parameter of the collision. (Impact parameter is a measure of how far off center the collision takes place. It is equal to the distance P in Fig. 2.)

Perform a number of collision experiments for different impact parameters, always using the same value of h to release the incident ball. Record the landing points using carbon paper on the same sheet of paper which you


 


                                                                  Figure 3

As you do this remember to

·        Label corresponding landing points (get five points for each impact parameter) as they appear so that you can identify them by collision and by which ball made them.

·        Adjust the position of the supporting screw for each new impact parameter so that the incident sphere makes contact in the same position when it leaves the ramp.

Before coming to lab try to predict if the points where the spheres hit the floor (the endpoints of the velocity vectors) will form any particular pattern. Do you observe any pattern?

For a few data pairs (i.e., the landing points of the two balls) draw the momentum vectors after the collision and compare them with the momentum vector before the collision.

To know the correct velocity vectors you must know the position of each of the two balls at the moment of impact. The plumb bob indicates the position of the target ball at the moment of collision. The position of the incident ball may be found by following Fig. 2. If you should decide to ignore the differences in position or to use some other approximation, justify your action in terms of the uncertainties involved.

                    Figure 4

Do your data indicate that, within experimental uncertainties, both energy and momentum are conserved for these collisions? If not, explain any discrepancies.

Part III: Collisions between Unequal Mass Balls

In this part we will investigate collisions between spheres of equal size, but unequal masses. The mass of the steel sphere, M, is obviously larger than that of the glass sphere, m. Which one should you use as the incident sphere? Follow the same procedure as in Part II. Record the landing positions of the two spheres after collisions with different impact parameters. To avoid confusion, use a new sheet of paper to record your data. Be sure to label the landing points after each experiment and record the point at which the impact spheres leave the ramp. Determine the masses of the two spheres you are using with a balance and record for later use. Whenever you take data be sure to record the uncertainty in your measurements. For example, in doing the collision experiments you should repeat the experiment five times for the same impact parameters to see how much your landing points scatter.

Before coming to lab try to predict how the locus of the endpoints of the velocity vectors will differ from those observed in the collisions of equal mass balls. Is momentum conserved within the experimental uncertainty of your measurement? Explain. Make a similar computation for the energies of the spheres before and after collisions. To what extent is energy conserved?

Computer Analysis

This lab uses Excel to simplify your calculations and to create your graphs. This greatly reduces the time required for you to complete this lab. In addition you will also get more practice using Excel.

Startup the Computer and Run Excel

Fill in your names, the date, and the section number in cells at the top of Sheet1.

Entering basic data about the collision

1.      Enter the height s of the launching point above the floor, the height h of the starting point on the ramp above the launching point, the mass m1 of the incident ball, and mass m2 of the target ball.

  1. Enter the estimated uncertainties in these quantities.

Entering the data for no collision

  1. In a column on your spreadsheet enter your measurements of the distances to the initial impact point from the location of the plumb bob. These are your measured values of D (see Figure 1).
  2. At the bottom of that column, compute the average value of D and the standard deviation of your measurements. Be sure to clearly label things in the column just to the left of the column containing your numbers.  Be sure that the standard deviation you get for D is consistent with the spread of values you see on the white paper.
  3. Now compute on the spreadsheet the value of the initial velocity in two different ways: first, by using the height s of the support and the horizontal distance D, and second, by using the height h and conservation of energy. Using propagation of errors, also estimate the uncertainties in these values.
  4. How well do these independent measurements of V0 agree? Discuss any reasons you can think of for the differences. Be sure to use the value determined using D in the subsequent analysis.
  5. Finally, compute the initial momentum of the incident ball.

Entering locations of balls after collisions

Enter your measured x- and y-components for the distances to each group of hits made for each impact parameter. For each impact parameter, you will enter two pairs of (x,y) values, i.e. one pair each for the incident and the target balls. You do not need to enter separate distances for each spot, just for each group of spots. The x-axis is the direction the ball goes if there is no target. The y-axis is perpendicular to that. If you enter positive numbers for all the y values, then in the later analysis remember to subtract the momenta in the y-direction from each other. Make sure you record the data for each ball consistently with their masses (keeping track of the subscripts 1 for incident and 2 for target).

Graphing the data

1.      In four columns, compute the components of momentum, Px1, Py1, Px2, and Py2 for each ball (incident and target).

2.      Make xy scatter plot graphs of Px1 vs. Py1 and Px2 vs. Py2 (make two graphs).

  1. If you entered all y-values as positive, and your x-axis was positioned correctly, then all the data in both plots for the equal mass case should lie on the same semicircle. If your x-axis was not positioned at y = 0 everywhere, you may notice two semicircles of slightly different radii. What do you see for your data?
  2. In the unequal mass case, you should also note that the relative sizes of the ball 1 and ball 2 circles are not the same in these graphs as they were on your raw data sheets. On these graphs you are plotting momenta. On the raw data sheets are plotted distances, which are directly proportional to velocities. Why is there a change in scale for the unequal mass ball plot?

Analyzing your results

1.      Compute a column containing Px1 + Px2 and compare the value with the initial momentum of the ball.. Compute a column containing Py1 – Py2. What do you expect for this column? How close did your data come?

2.      The sum of the angles measured from the x-axis should be 90° . Compute a column containing theta1 + theta2? How close did your data come?

  1. Compute a column containing the total kinetic energy after the collision. If these were elastic collisions, the total kinetic energies after the collision should equal the kinetic energy before the collision. How close were your data?
  2. You can also average your data for x-momentum and y-momentun, theta, and kinetic energy and get the standard deviations. Are your values and estimated uncertainties reasonable?

Now repeat the procedure for unequal mass balls. Your partner should do the typing and mouse moving this time. When finished, print out a copy of your spreadsheet to hand in with your lab report.

Some physics hints

1.      When the incident ball starts at rest and drops through a height h, it loses potential energy of the amount U = mgh, and gains kinetic energy of the amount K = ½ m V02.

2.      When a ball drops through a vertical height s, starting with an initial horizontal velocity V, it will spend t = sqrt(2s/g) seconds in the air, and will go a distance D = V t in the horizontal direction.

3.      The x- and y-distances you measure are therefore easily converted into x- and y-components of velocity (just divide Dx/t, etc), and multiplication by m then converts these velocity components into momentum components.