Abstract

The purpose of this lab is to verify the principle of conservation of momentum within the uncertainties of the experiment. The conservation of momentum lab helps the physics student understand how momentum is conserved between two or more particles during a collision sequence. To test the theory of conservation of momentum an air table with two circular masses of varying mass were collided together and video taped. The digital video was then analyzed to check for conservation of momentum. A second video was analyzed to determine the mass of each of the colliding circular objects. Momentum in this experiment was not conserved in the x direction within the range of uncertainty. The y component of the experiment did have conservation of momentum within the ranges of uncertainty.

Procedure

A blower was connected to an air table to negate the frictional force on the circular mass (puck) as it slid across the table colliding with a second puck. The type of air table used for this experiment is shown in Figure 1. Prior to conducting the experiment the mass of each puck was determined. Mass 1 is a blue puck with purple tape attached to its top with a mass of 50 grams at (+) or (-) one gram on the first scale and 49.8549 grams on the second scale at (+) or (-) 0.0001 gram. The second puck had a mass of 67 grams on the first scale and 66.5356 grams on the second scale with the same uncertainty as the first mass. The diameter of the small red puck is 7 cm (.07 m). The diameter of the large blue puck is 9 cm (.09 m).

Figure 1: Air Table and Pucks

The table was marked in 5 cm squares prior to the experiment to facilitate distance and angle measurements from the digital video. The red puck with the most mass was placed in the center of the table with the blue puck launched at the red puck so it would collide with the red puck. After several practice collisions the impact was recorded using a digital video camera. Several impacts were done with the best one saved for analysis. This impact is shown below.

Conservation of Momentum Experiment

Analysis

Before attempting to determine if momentum and/or energy was conserved in this collision system several areas of uncertainty arise. They are the distance and angle measurements derived from the scale drawn on the air tabletop, the mass determination of each puck and a lack of conservation of momentum by definition in this collision.

The scale on the tabletop is purported to be .005 m². It is obvious that the lines are not perfectly straight and are obscured in places. The chalk marks are several mm in width extending each block in each direction. When measuring the center of mass movement of the puck the best estimation is on the line or between the lines giving a measurement estimation of 2.5 centimeters. This is a large uncertainty introduced into the experiment. This uncertainty extends into the approach and departure angles of each puck.

The mass uncertainty is relatively small compared to that of the tabletop measurements when using the mass given from the second scale, which is to the 0.0001 grams for each mass and is shown below.

By definition momentum in this experiment was not conserved as indicated by the textbook, Physics For Scientists and Engineers, volume 1, page 254, “Whenever two or more particles in an isolated system interact, the total momentum of the system remains constant. This law tells us that the total momentum of an isolated system at all times equals its initial momentum. The only requirement is that the forces must be internal to the system.” When viewing the video, note that the red puck is redirected in the y direction by a wire surrounding the air table, a second external force acting on the red puck. Note that the blue puck is redirected in both the x and y direction back onto the air table by the “u” shaped apparatus in the lower right corner after being redirected by the wire surrounding the table. The wires forces that are acting on the pucks by redirecting their direction can be accounted for and ignored for the most part. This is because the departure angle of both pucks is clear at impact and is not affected by the wire redirection. The problem that does present itself is the amount of slowing by the wire and then possibly imparting an external spring force to the puck. As shown in the video the red puck does stop in the y direction and is redirected but continues in the x direction. The blue puck has the spring force of the wire redirecting it plus the apparatus attached to the table redirecting the puck. As seen in the video, this apparatus moves while the puck is in contact with it redirecting the blue puck back onto the air table. These are all external forces acting on the system slowing and redirecting the pucks or changing the momentum of the initial impact. An analysis of the system will be attempted to see if the momentum in the x or y direction is conserved or if the momentum can be determined.

It seems that the energy of the system may not be conserved due to the additional impacts of the pucks. The spring constant of the wire is unknown. The x movement may be estimated from the video but would be imprecise. The initial velocity of the red puck at the time it contacts the wire is considerably higher than the final velocity off the wire, indicating that the wire absorbs the energy during deformation of the wire. Although the energy of the red puck may be determined by the relationship of energy conservation and Hooks Law, , the blue puck has struck part of the table absorbing the pucks energy without any deformation to the table or puck making an analysis of its post impact energy very difficult if not nearly impossible for this experiment.

Several steps went into analysis of the pre and post impact motions of the two pucks to see if momentum was conserved. The speed of the video play back; the release point of the blue puck; the impact point between the two pucks; the approach and departure angle of the pucks and the final resting point of each puck was determined. This analysis allowed for the calculation of the pre and post impact velocity for each puck. On most calculations all non significant figures were held until the end of the calculations.

It was determined that the digital camera recorded the event at 24 frames per second with a total of 134 frames or 5.40 seconds. Impact occurred at frame 64 and final rest occurred at frame 134 for the blue puck and frame 97 for the red puck. The release point for the blue puck was at frame 61 when all possible acceleration of the puck was completed. The times for each event is 0.125 seconds for the blue puck from release to impact; 1.375 seconds for the red puck to come to rest; and 2.92 seconds for the blue puck to come to rest.

The pre-impact travel distance for the blue puck was approximately 10 cm in the x direction and 2.5 cm in the y direction for a total distance of 0.103 m using . Dividing the travel distance d by the time t yielded a

pre-impact velocity for the blue puck, . The red puck was at rest at impact. The post impact travel distance of the red puck including the bounce off the wire was 74.24 cm. The blue puck had a post impact travel distance of 74.33 cm. The post impact velocity of the red and blue pucks were calculated in the same manner yielding a post impact of v_Rf=0.533997 m/s for the red puck and v_Bf=0.2545559 m/s.

The approach angle for the blue puck was determined by the pucks position at acceleration release compared to its position at impact. The pucks approximate center of mass was used to determine all positions of each puck on the air table. The coordinates (0,0) were set at the lower left corner of the air table with the wire surrounding the table extending from the origin as the x and y axis. The blue puck started at a position of (20,15) and ended at the coordinates of (30,17.5). The approach angle was determined by . The departure angle of the red buck was determined using the same type of analysis. Its rest position was at coordinates (32.5,25) and was redirected at coordinates (50,47.5). The angle was determined by the relationship . At impact, the blue puck had the coordinates of (30,17.5) and was redirected at (60,2.5). The blue pucks post impact departure angle was . Setting the approach angle to 0 degrees and rounding to two significant figures for the angle translated the departure angles of each puck to 38 degrees and 40 degrees. The angle between the two remains at 78 degrees.

The initial and final velocities were calculated by the formula in all cases. The distances were derived from the board markings and diameter of the pucks, which has introduced a sizeable error into the equations. At (+) or (-) 2.5 centimeters per measurement the uncertainty starts to compound for the velocity calculations. The initial velocity for the blue puck was calculated from the initial coordinates of (20,35) and impact coordinates of (30,37.5). The total travel distance of the blue puck pre-impact was . The time to travel the 0.103 m was over three frames or . The initial velocity is . The initial velocity uncertainty is and .

The post impact velocity of the red puck was calculated using the same method described above. The starting coordinates were (32.5,25) and the ending coordinates were (52.5,50). There is no y coordinate of 52.5, this is from the redirection of the puck of 5 cm at the wire or 47.5 (-2.5)= 50. The time was over 33 frames or 1.375 seconds. The total distance for the red puck from start to rest was . The total distance traveled is . The post impact velocity for the red puck is 0.23284 m/s. The uncertainty for the red puck is . The time uncertainty is 0.03 seconds.

The blue puck traveled further than the red puck and was redirected a considerable distance after encountering two additional impacts. The total travel distance from impact to rest was 0.7433 m in a time of 2.92 seconds. The uncertainty for the distance is 0.0033 meters and the time uncertainty for the blue puck is 0.01 seconds.

Conservation of momentum is or . The initial velocity of mass 2 is zero.

The momentum in the x direction is

As predicted the momentum in the x direction is not conserved for this experiment. The momentum difference P=0.019 kg m/s is not even close to showing conservation in the x direction. The uncertainties on each side of the equation do not account for the large discrepancy for the momentum values. The only way to account for the momentum not being conserved in this impact is the external forces acting on the pucks. The y direction values should show some discrepancy relating to the secondary impacts.

With the initial angle set at zero there is no initial momentum in the y direction therefore: . The negative sign indicates that mass one has a downward y component. The y direction momentum calculations are:

This value is within the uncertainty for the measurements off the video and does indicate that the momentum was conserved for the y component of this impact. But overall the momentum was not conserved.

The second section of the experiment calls for deriving the mass ratio of the particles shown in the strobe video.

Strobe Movie

The mass ratio for kinetic energy is: (units are in pixels)

Assuming momentum is conserved the mass of the particles are: (units are in pixels)