FVCC Physics I Laboratory. Inelastic and Elastic Collisions

Transcription

1 FVCC Physics I Laboratory Inelastic and Elastic Collisions J.K. Boger September 18, Objective To observe and take data on both inelastic and nearly elastic collisions. Test the law of conservation of linear momentum by experiment. 1 2 Theory 2.1 Collisions Momentum is the concept of inertia. It is given by equation 1. p = m v (1) This definition leads to a new form for Newton s second law given by equation 2. F = d p dt Equation 2 is in fact a more accurate way to state Newton s second law since it allows for the possibility of a changing mass. In our experiment we will only deal with a changing velocity and not a changing jboger@fvcc.edu 1 This exercise originated with R. Schaus and this is an adaptation of his work (2) 1

2 Physics 1 Laboratory Spring Semester 2 mass. In addition to Newton s second law, the third law plays an incredibly important part in the concept of conservation of linear momentum. The third law requires us to isolate systems with Free Body Diagrams (FBD), breaking the often inconvenient action-reaction force pairs. When the FBD isolates a body or system such that the net force is zero, then a consequence is the conservation of linear momentum. This is stated mathematically in equation 3. F net = 0 = d p dt Here the change in momentum is clearly the quantity which is zero, as opposed to the changing time. We could integrate both sides of equation 3 to yield the simpler version of conservation of momentum given in equation 8. p = 0 (4) In a before and after (collision) sense this equation is written as equation 5. (3) p f = p i (5) This equation is valid as long as there is no external forces on the system. If a small amount of friction is acting on the carts used in our experiment, then equation 5 will not hold exactly, but it will be close. Writing out equation 5 in terms of velocity exposes the quantity we will measure in this lab, velocity. Further, we have removed the vector designation since all the motion studied in this lab is along one axis which we select as the x-axis. m 1 v 1xf + m 2 v 2xf = m 1 v 1xi + m 2 v 2xi (6)

3 Physics 1 Laboratory Spring Semester 3 Here the i subscript stands for initial while f stands for final. Equation 6 is the principle equation use in this experiment. Equation 6 works perfectly for inelastic collisions where the final velocity of cart 1 and cart 2 are the same since they are stuck together. But this does not happen for elastic collisions. Instead the carts bounce off each other and each as a unique final velocity. Mathematically this means equation 6 is not enough since there will be 2 unknowns. For this case we have to turn to the second major conservation principle in mechanics: conservation mechanical of energy. When applied to the carts colliding on a level platform, conservation of mechanical is given in equation 7. Naturally the common factor of 1/2 could be divided out of this equation. 1 2 m 1v 2 1xf m 2v 2 2xf = 1 2 m 1v 2 1xi m 2v 2 2xi (7) Typically, we can use equations 6 and 7 to find the final velocities of the two carts given the initial velocities. But it is possible to consider a special case where the initial velocity of mass 2 is always zero. Even if mass 2 is moving, we can always use Galilean relativity to transform into the moving frame of mass 2 and thereby again be in the condition of v 2xi = 0. In this case, the conservation of momentum is written by equation 8 and conservation of energy simplifies to equation 9. m 1 v 1xf + m 2 v 2xf = m 1 v 1xi (8) m 1 v 2 1xf + m 2 v 2 2xf = m 1 v 2 1xi (9) Equations 8 and 9 can now be solve for the relativistic 2 velocities after an elastic collision. v 1xf = m 1 m 2 m 1 + m 2 v 1xi (10) 2.2 Errant External Forces v 2xf = 2m 1 m 1 + m 2 v 1xi (11) If friction is present, as it almost always is, it will contribute to an error in equation 6. Going back to Newton s second law we can write an equation that captures friction. This is given in equation 12. p = µmgt (12) Further, if the experiment is not level, a small component of gravity, often given by F g = mgtsin(θ), would also act to interfere with our conservation experiment. Equation 13 adds or subtracts (depending on the slope) to the friction force. 2 Note that this is not to be confused with Einstein s theory. p = µmgt ± mgtsin(θ) (13)

4 Physics 1 Laboratory Spring Semester 4 In equations 12 and 13, t represents the time between the before and after of the experiment. This is powerful information. It tells us what to expect for potential systematic error in our experiment testing the concept of conservation of energy. The simplest way to reduce this error is to limit the time of the experiment given by t in equation 13. Other ideas are to check the level of the experiment, and try to limit the amount of friction. But further, this equation exposes a way to explore the errors. For example, imagine working to level the track in the experiment so as to eliminate the gravity error and isolate the friction. Simply by recording change in momentum for various experimental times would provide a data set which could be plotted in a p vs. t plot, the slope of which is the µmg term. Producing graphs of data is a great way to average data and thus measure the target quantity, µ in this case. 3 Procedure 3.1 Measure the Inelastic Collision 1. Choose the carts with velcro tabs on each and ensure they do not repel each other. (Half the carts have magnets in the ends and will repel other carts.) 2. Measure the mass of each cart and record these values. Note, M is the uncertainty in mass. 3. Place the Vernier motion detector at the end of the track such that M 1 move away from the sensor as it is pushed down the track. 4. Set-up the Vernier LabQuest to take data for 3 seconds, and 20 samples per second. 5. With M 2 at rest, start the data collection and gently shove M 1 down the track towards M 2. The motion detector should record the velocity of M 1 and then M 1 + M 2 together. 6. Repeat the data as many times as necessary until you feel you have good clean data. 7. Given the measured initial velocity, calculate the theoretical value for the final velocity. 8. Compare your calculated final velocity to that measured.

5 Physics 1 Laboratory Spring Semester Measure the Nearly Elastic Collision Case I: M 1 = M 2 1. Choose the carts with repellent magnets in them. 2. Measure the mass of each cart and record these values. 3. Place a Vernier motion detector at each end of the track. One sensor should keep track of M 1 and the other will track M If the sensors interfere with each other, then measure M 1 with the motion detector and use a photogate to measure the velocity of M 2 5. Repeat experiment until you are satisfied with the data, then plug LabQuest into the computer and open LoggerLite. Data can be exported to Excel. 6. Analyze the data for velocities. 7. Compare the measured velocities to those calculated by equations 10 & Estimate errors as appropriate Case II: M 1 > M 2 1. Choose the carts with repellent magnets in them. 2. Measure the mass of each cart and record these values. This should include added mass to M 1 3. Place a Vernier motion detector at each end of the track. One sensor should keep track of M 1 and the other will track M If the sensors interfere with each other, then measure M 1 with the motion detector and use a photogate to measure the velocity of M 2 5. Repeat experiment until you are satisfied with the data, then plug LabQuest into the computer and open LoggerLite. Data can be exported to Excel.

6 Physics 1 Laboratory Spring Semester 6 6. Analyze the data for velocities. 7. Compare the measured velocities to those calculated by equations 10 & Estimate errors as appropriate Case III: M 1 < M 2 1. Choose the carts with repellent magnets in them. 2. Measure the mass of each cart and record these values. This should include extra mass for M Place a Vernier motion detector at each end of the track. One sensor should keep track of M 1 and the other will track M If the sensors interfere with each other, then measure M 1 with the motion detector and use a photogate to measure the velocity of M 2 5. Repeat experiment until you are satisfied with the data, then plug LabQuest into the computer and open LoggerLite. Data can be exported to Excel. 6. Analyze the data for velocities. 7. Compare the measured velocities to those calculated by equations 10 & Estimate errors as appropriate 3.3 Reporting results 1 Be sure to include the plots from Vernier data in your report, annotated to illustrate which points on the plots are used for calculations. 2 Considering the discussion on errant forces (section 2.2), can you estimate the rolling friction from your motion data? Revision date: September 18, 2013