Experiment 6: Collisions on Air-Track
Purpose (1) To study the Principles of Conservation of Momentum and Conservation of Mechanical Energy.
Apparatus Air-track with sparker and supplies (recording tape, sticking tape, wooden sticks, etc.) Equal-arm balances (on the side)
Theory EQUATION (1) (A) Conservation of Momentum In a collision of two objects (like the air-track “riders”), their moments are changed. However, M Ui + M Ui = the Total Momentum (of both objects) is conserved, A A B B as in equation 1) where: M Uf + M Uf MA, MB = masses of riders A A B B UiA, UiB= their velocities before the collision UfA, UfB = their velocities after the collision
(B) Conservation of Mechanical Energy If the internal forces between the colliding objects EQUATION (2) are conservative, then the Total Mechanical 2 2 Energy will be also conserved. If only the kinetic ½ MA UiA + ½ MB UiB energy is involved, an equation like (2) results. = Collisions where the Total Kinetic Energy is 2 2 ½ MA UfA + ½ MB UfB conserved are called “elastic collisions”.
(C) Coefficient of Restitution If the internal forces between the colliding objects are dissipative (at least partially). EQUATION (3) the Total Kinetic Energy is not conserved, resulting in inelastic collisions. To compare elastic and inelastic ( ) e = - UfB-UfA collisions, it is convenient to quote the Coefficient of Restitution “e” given by (3). In elastic collisions e = 1. (U U ) If there is a loss of Total Kinetic Energy, then e 1. iB- iA The greatest possible loss occurs when e = 0 – the colliding objects stick to each other and move together after colliding.
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(D)Air-Track: This is a device allowing sliding objects (“riders”) to move along the track supported by an air-cushion (resulting in negligible friction). To measure the speed of a rider, a timing device (“sparker”) is used. Your instructor will explain the operation of these devices. Caution: Do not attempt to operate the air-track and/or the sparker without instruction; the high voltage produced by the sparker is dangerous.
Do not move the riders along the track unless the air-supply is on: the track is easily damaged by scratches, and it is expensive.
Procedure Preliminary Part: Testing the Air-Track (a) With the air-supply “on”, place a rider, gently, on the track and release it from rest: the rider should remain motionless. If the rider moves, the track is not horizontal: call your instructor (or the technician). Do not proceed further until the track is adjusted. (b) With the air-supply “on”, place a rider on the track and give it a moderate push: it should bounce many times off the ends of the track before appreciably slowing down (or stopping). Test all riders this way. If this test fails, it means that there is too much friction: call your instructor (or the technician). Do not proceed further until everything is adjusted. Warning: Every time a rider jumps off the track or is dropped, it may be very slightly bent: you will never notice this, but it may fail to pass the above test. (c) Make sure the sparker is operating properly when the rider is at rest (adjust the pliable wire on the rider). Attach the recording tape, glossy side up, about 50 cm long, in the middle of the track. Placing the rider near the left end of the track, hold it with the wooden stick, while your partner starts the sparker. Give the rider a moderate push to the right, creating a “test-run” at constant speed. Your tape record should look like this:
......
Fig. 1 The Record of a Uniform Motion
Important: (i) the dots on the tape should be sharp (not smeared) (ii) no dots should be “missing”
*****DO NOT PROCEED FURTHER UNTIL YOUR INSTRUCTOR APPROVES YOUR TEST-RUN.
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Procedure Part I: Elastic Collisions
(d) Use the heavier rider as “rider A” : it should have the pliable wire and a holder for additional mass attached to it. Measure and record (under Run #1) its mass as MA. Also, record the mass MB of the “rider B”. (e) Attach the recording tape (about 100 cm in length, in the middle of the track). Place the “rider A” near the left end of the track and the “rider B” about 20 cm to the right of the midpoint of the tape. Holding the “rider A” with wooden stick, with air-supply “on”, start the sparker. Give the “rider A” a moderate push to the right to achieve the collision. Keep the sparker “on” after the collision and stop it just before the “rider B”, rebounding from the right end of the track, collides again with the “rider A”.
Important: To achieve good accuracy, make sure that UA is at least 30 cm/sec. but avoid excessive speeds: if the rider jumps off the track, it may be fouled up: see the warning under (2).
*****CHECK YOUR TAPE RECORD FOR QUALITY. IF IN DOUBT, CALL THE INSTRUCTOR. (f) If the tape is good, make additional runs for all other partners at your air-track. All students must immediately label (as Run #1) and sign their own tapes. (g) Increase the mass of “rider A” (check with your instructor for details) and repeat and (f) as “run #2”. Record the new mass.
Procedure Part II: Inelastic Collisions
(h) Use a new “rider A”, so that riders “A” and “B” have velcro on the ends facing each other (the elastic bands on their other ends). Measure and record MA and MB, as before, under “Run #3”, (make sure MA includes the pliable wire and the holder for additional masses). (i) Follow the instructions under (e), with obvious modifications (riders “A” and “B” will now stick together after the collision). Make additional runs, as per (f). (j) Increase the mass of “rider A” (check with your instructor for details) and repeat (h) and (i) as “Run #4.
Before you leave the Lab: All your (personal) tapes, properly labeled, and signed with your name, must be initialed by your instructor along with your data-sheet. Also, make sure you understand how to process the data.
Lab Report 31 Experiment 6
Part I:
Note: You must attach your tapes to your report: fold them in a length of about 10 cm, secure with a paper clip, put them in an envelope, and staple the envelope to your report.
(1) Your tape should look like this: Collision Region
......
d d ‘
Fig. 2 The Record of a Collision
Ignore the “collision region” and other useless portions of your tape (perhaps, at the left and right ends: consult you instructor). Make the “marks” (as in Fig. 2), selecting as large intervals d and d1 as (safely) available and quote (both on the tape and in the report) the number N of the intervals (Important: If an occasional dot is missing, account for it). Measure d and d1 and calculate UA and VA (remember that the sparker produces one dot every 1/50 of a second).
(2) Draw and fill out Table #1. Note: You need not show the details of your calculations, but observe the following (i) UiA and UfA are obtained from your tapes. (ii) UfB follows from (1). (iii) Appropriate units must be used and quoted. (iv) % loss in KE (Kinetic Energy) should be calculated with respect to the “Total KE before”. If there is a measured gain in KE, enter it with minus sign.
TABLE # 1. ELASTIC COLLISIONS RUN M M Ui Uf Uf e Total KE Total KE % Loss # A B A A B Before After in KE 1 2
(3) Comment on how well your results confirm the Principle of Conservation of Total Kinetic Energy (if your discrepancies are large, locate you blunders and correct them before you submit your report).
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Part II:
(4) Draw and fill out Table #2.
TABLE #2. INELASTIC COLLISIONS. Measured Theoretical % RUN MA MB From Tape Uf = MA . Ui Discrepancy A A # UiA UfA in UfA MA + MB 3 4
(5) Comment on how well your results confirm the Principle of Conservation of Total Momentum. (If you discover blunders, correct them before you submit the report.) (6) Draw and fill out Table #3. Note: Use the measured values of VA here.
Table #3. INELASTIC COLLISIONS. RUN Total KE Total KE % Loss # M M Ui Uf е A B A A Before. . After in KE 3 4
Questions
(1) What are the theoretical values of the % -loss in Total Kinetic Energy in runs #3 and #4. (2) If a rubber ball is dropped from the height of 120 cm from the floor, and rebounds to a height of 80 cm, what is the value of e? Show your reasoning. (3) Can you conceive of a situation where the (internal) forces between the colliding objects are conservative, but the collision is still inelastic (total kinetic energy is not conserved)? If you do, give a practical example.
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