Collisions. Objective. Introduction

Transcription

1 Collisions Objective To study conservation of momentum and kinetic energy during a collision between two bodies. Introduction In this experiment a moving air puck makes a glancing collision with a puck that is at rest. The objective is to compare the kinetic energy and the momentum of the system before and after the collision. This seems a simple task because the frictionless pucks are expected to move with constant speed on the flat table with the timer marking their positions at regular intervals dt. In practice, the measurements must be exercised with care if energy and momenta are to be precisely determined. Some of the complexities which mask the simple principles are outlined in the description below. The use of the air table and timer are described in Linear Acceleration and Projectile Path. Two sets of pucks are provided, one of them magnetic. The experiment is to be carried out once with each set. NOTE: Each student should make her/his own set of tracings and calculations. Otherwise the workload ends up being uneven and you won t learn as much. Theory There are only two relevant principles to this lab. The first is universal: the Law of Conservation of Momentum. This law simply states that the TOTAL momentum (P) of a system before and after any event (which in this case, is a collision) must remain constant. For two masses (pucks) m 1 and m 2, this can be written as follows:

2 2 P = P, p 1 + p 2 = p 1 + p 2 m 1 v 1 + m 2 v 2 = m 1 v 1 + m 2 v 2 where v 1 and v 2 are the velocities of m 1 and m 2 respectively BEFORE the collision, and v 1 and v 2 denote the velocities of each mass AFTER the collision. Notice that the v s and p s are written in BOLD, indicating that they are vectors. Momentum is a vector quantity, encompassing both magnitude and direction. It s important to keep track of the directions, or you ll become very confused very quickly. The second principle we will investigate is the Conservation of Kinetic Energy. A collision in which the total kinetic energy before and after the collision remains unchanged is described mathematically by: p 1 2 KE = KE, 2 + p 2 = p 1 + p 2 2m 1 2m 2 2m 1 2m m 1v m 2v 2 2 = 1 2 m 1v m 2 2v This type of collision is referred to as an elastic collision. Most tangible objects you re familiar with (such as pickup trucks, baseballs and air pucks) usually do NOT undergo elastic collisions. In general, some energy is lost as heat or sound during macroscopic collisions, with some exceptions. Therefore, you shouldn t expect a collision between two steel pucks to be elastic. Note that this doesn t mean that total energy is not conserved! It simply means that kinetic energy is not conserved, as some of it is converted into a form which you can t measure. Total energy is always conserved - this is one of the most fundamental concepts in physics. Notice that that the velocities and momenta in the above kinetic energy equations are NOT written in bold, even though you know they are vector quantities. This is because we

3 3 are referring only to the magnitude of the vector, not its direction. It s a pretty standard convention in physics to write v (or v) when we are talking about both the size and direction of the velocity, and to simply write v when we re only interested in the size (and not the direction). Since it s hard to write in bold in your log book, we suggest you use arrows above the letters (eg. v, p) when referring to vectors, and omitting the arrows when you re referring to scalars. Procedure To simplify the analysis, we ll start with one of the pucks (say, puck #2) at rest. This means that v 2 (and therefore p 2 ) will be zero. Carefully level the table and smooth the paper on it so there are no bulges. An ideal trace will have the same number of points in the incoming path as in each of the exit paths. Make a few trial runs to get a feel for the best way to produce the collision. Make sure that neither puck moves too slowly, but don t make the collision too violent. The stationary puck should be prevented from moving before the collision, to the best of your ability. The air hoses have a tendency to tangle and stretch, so try to eliminate this completely to get good results. In a clean collision the steel pucks interact with a sharp click (not a thud) and the magnetic pucks are silent. Pick a suitable timer setting dt that will give make a good density of sparks (don t forget to record dt and its uncertainty!) and produce a trace for a glancing collision of the steel pucks. Record the masses of the pucks, noting which one was initially at rest. Repeat this procedure for a glancing collision of the magnetic pucks (which are lighter and have a plastic extension). Note that the paths are curved in the intersection region because of the long-range magnetic forces.

4 4 Analysis With straight edge and pencil draw the straight line incoming and outgoing paths for each puck by connecting the appropriate spark points. Taking the initial path (i.e. along the direction of v 1 ) as the x-axis, measure and record the angle each of the final paths (v 1 and v 2 ) makes with the x-axis. An example experiment with the steel pucks is illustrated in figure 1. Calculate the the speeds immediately before and after collision by measuring the displacement in a few time intervals. You re interested in what s happening just before and just after the collision. Therefore, it s not a great idea to use ALL the points in the trace for these measurements; you can t be certain that the pucks were completely unaffected by outside forces (such as friction, air drafts and clumsy lab partners). Such disturbances could cause a change in the velocities, and therefore might throw off your conclusions. However, taking too few points may increase your measurement uncertainty (due to simple statistical rules). Use your best judgement as to how much or little of the trace to take. You can now compute both the initial and final kinetic energies, and also the x and y components of the linear momentum before and after the collision. For the magnetic pucks determining the incoming and outgoing tracks is more difficult. You will have use points far away from the point of impact in order to make a reasonable estimate of the velocities. It may be appropriate to use an increased margin of error for this analysis. m2 m1 FIG. 1: Example of producing a collision trace with the spark table. Note that in the lab the spark points are produced on the UNDERSIDE of the page, and won t appear on top as illustrated here.

5 5 Name: Student ID: Partners Name: Date: WORKSHEET: To be handed in: 1. The following table (note: the table is continued on the next page): STEEL PUCKS MAGNETIC PUCKS VALUE UNCERTAINTY VALUE UNCERTAINTY dt θ 1 θ 2 v 1 v 2 v 1 v 2 (v 1 ) x (v 1 ) y (v 2 ) x (v 2 ) y m 1 m 2 KE 1 KE 1 KE 2 KE 2

6 6 STEEL PUCKS MAGNETIC PUCKS VALUE UNCERTAINTY VALUE UNCERTAINTY (p 1 ) x (p 1 ) y (p 1 ) x (p 1 ) y (p 2 ) x (p 2 ) y (p 2 ) x (p 2 ) y 2. Sample calculations (including the corresponding uncertainty calculation) for (v 1) x, KE 1, (p 1) x. You only need to show the calculations for the Steel Pucks. 3. Within the limits imposed by your uncertainties, what does your data permit you to conclude about the collision? Is kinetic energy conserved? What about linear momentum? 4. Comment on any differences between the outcome for the magnetic pucks and the steel pucks. 5. What if you had mixed up which trail was incoming and which was outgoing? In other words, if you had labelled the trails after the collision as being before the collision by mistake, and subsequently carried through the analysis, would your conclusions be the same?