Lab Report 4_ Impulse and Momentum

Verifying the Impulse-Momentum Theorem Equation PCS 120

Introduction The objective of this experiment is to verify the impulse-momentum theorem which is defined as the following, , the theorem relates two systems together during a 𝐽 → = 𝐹 → ∆𝑡 = ? ? 𝑣 ? − ? 𝑖 𝑣 𝑖 collision. In order to prove the theorem it must be verified with experimental results that agree with the theory such that impulse over momentum should have a slope of 1. The data collection of the investigation will be recorded using the Vernier Force Sensor on the post and the Vernier Motion Sensor at the end of the aluminium track which is equipped to a string and elastic band connected to a wooden block with varying mass configurations, and the software Vernier Graphical Analysis will provide two important graphs with a force-versus-time graph and a velocity-versus-time graph. Theory In a collision an object approaches with an initial velocity, experiences a force over the duration of the beginning of contact to the end of contact and then leaves with a final velocity, the force experienced over the duration is more specifically an impulsive force which is a large force exerted on an object over a small interval of time (Knight, 2016). Impulse ( ) is the quantified as 𝐽 → the effect of the force acting on an object over time in which an external force results in change a in momentum, consequently, impulse can be calculated as and given that impulse 𝐽 𝑥 = 𝐹 𝑎𝑣? ∆𝑡 is equal to the change in momentum another relationship can be described using the momentum principle, (Knight, 2016). Momentum is the product of an object’s mass and velocity, ∆𝑝 → = 𝐽 → which can be written as . Impulse and momentum are directly related to one another ∆𝑝 → = ?𝑣 → as stated in the momentum principle and is also evident when comparing the concepts graphically as in Figure 2 and Figure 3 , the area under the curve of a force-versus-time graph is equivalent to the impulse which is equivalent to the translated velocity-versus-time graph in which the section between where the contact begins (v ix ) to where the contract ends (v fx ) is the duration in which impulse changes the momentum.

Procedure This tutorial helps to perform and collect data to verify the impulse-momentum theorem equation, by calculating values necessary to confirm the theorem. The trials include accounting for velocity, total duration of collision, final velocity, initial velocity and mass to ensure impulse and momentum can be calculated. Ensure to have the Vernier Graphical Analysis software downloaded on a computer along with a Vernier Force sensor with a post and a Vernier Motion sensor set up on the other end of a levelled aluminium track, the force sensor must be hooked up the vernier cart by an ‘S’ hook which is attached to an elastic band and a string with a total of six 500g ± 1% masses. Make sure the Vernier Graphical Analysis software is launched and connect the force sensor and set the range to ± 10N on the sensor. The force sensor is calibrated by performing a two-point calibration using a 0 g mass and a 500 g mass. Click the Sensor Meter button in the bottom right and click calibrate. Assure the force sensor is held vertically and enter the known value as 0 and click “keep”. In the second part of the calibration the 500 g mass hangs from the sensor, the second known value, 4.91N is entered and clicking keep saves the data. Click “apply” to complete the force sensor calibration. Place the vernier cart with the attached string and elastic band to the aluminium track, there are a total of 7 mass configurations and an additional 3 trials with varying initial velocities. For the first trial place the vernier cart on the track, ensure the force sensor is hooked onto the string and the force sensor is levelled to the vernier cart; press collect on the Graphical Analysis software and push the vernier cart forward and wait for the cart to experience impulse and return, shortly after the trial is complete keep the run and rename it accordingly to the trial number and amount of mass used. The Graphical Analysis software will produce two graphs, the resultant graphs on the software should be a force-versus-time graph and a corresponding velocity-versus-time graph. On the force-versus-time graph select the area under the curve as shown in Figure 1 starting from the beginning of contact to the end of contact and use the “graphing tools” option in the bottom left to see the statistics of the selected portion and take note of the mean force (N), the initial time and the final time. From the velocity-versus-time graph select the areas of constant velocity points and take note of the initial velocity value and the final velocity. There are a total of 10 trials with 3 trials of varying initial velocity and 7 trials with the varying 7 mass configurations possible and each configuration will include the vernier cart. The 10 trials of the experiment include (1) vernier cart, (2) vernier cart + 500g, (3) vernier cart + two 500g, (4) vernier cart + three 500g, (5) vernier cart + four 500g, (6) vernier cart + five 500g, (7) vernier cart + six 500g, (8) vernier cart + four 500g + light push, (9) vernier cart + four 500g + medium push, (10) vernier cart + four 500g + hard push.

At the end of the data collection, access the File menu in the top right corner to save and export the file as a gambl so that the file can be opened on the software, also download the data as a CVS so that it can be later downloaded as an excel sheet, where necessary analysis can take place to calculate the impulse and momentum which will be used to conclude experimental results with respect to the impulse-momentum theorem. Results and Calculation The impulse-momentum theorem states that impulse is equal to the change in momentum. If this is true, then a graph showing impulse vs momentum should have a slope of 1, proving they are equal to each other. First, the basic values are needed to continue with any calculations. The same calculations have been used throughout the results section, and examples of how each has been used are shown below: Calculating mass F G = mg 4.855 = m(9.81) m = 0.495 ±0.006 kg Determining Momentum: Mass = 0.595kg p = m f v f - m i v i p= 0.595( 0.584 - (-0.777) ) p = 0.809795 ±0.035 kg*m/s Determining Impulse: Trial 3 J = FΔt J = 1.321(0.6) J = 0.7926 ±0.31 N*s Since the values received from conducting the experiment are not ideal (unaccounted forces and small errors in measurements), there must be uncertainties and errors to be considered. Each of these calculations has been repeated for every value used in the creation of the overall graph. Velocity uncertainty: *σ vf and σ vi were calculated by taking the standard deviation of each v f and v i value* σ V =√(σ vf ² + σ vi ²) σ V = √(0.21658716² + 0.26623133²) σ V = 0.05766229 m/s σ V = ±0.058 m/s Momentum uncertainty: Trial 3 * v = v f - v i σ p /p =√((σ m /m)² + (σ v /v)²) σ p = p •√((σ m /m)² + (σ v /v)²) σ p = 0.809795 •√((0.00599187/0.595)² + (0.05766229/1.361)²) σ p = ±0.035264922 kg * m/s σ p = ±0.035 kg * m/s Mass uncertainty: Trial 3 σ MC is the uncertainty of the mass (±0.01%) σ MW is the uncertainty of the masses added on(±0.01%) σ m =√(σ M ² + σ MW ²+ σ MW ²) σ m =√(0.00595)² + (0.0005)² + (0.0005)²) σ m = 0.00599187 σ m = ±0.006 kg Impulse uncertainty: Trial 3 σ J /J =√((σ F /F)² + (σ t /t)²) σ J =J•√((σ F /F)² + (σ t /t)²) σ J =0.7926•√((0.507894663/1.321)² + (0.045946829/0.6)²) σ J =±0.310722531 N*s σ J =±0.31N*s

References Knight, R. (2016, January 4). Physics for Scientists and Engineers: A Strategic Approach with Modern Physics (4th ed.). Pearson. Appendices Trial F (N) t 1 (s) t 2 (s) t(from ∆ F>0) m (g) V f (m/s) V i (m/s) 1. Cart (495g) 1.469 N 0.8 1.3 0.5s 495 0.695 -0.879 2. Cart (495g) + 1(50g) 1.488 N 0.95 1.5 0.55s 545 0.711 -0.878 3. Cart (495g) + 2(50g) 1.321 N 0.7 1.3 0.6 595 0.584 -0.777 4. Cart (495g) + 3(50g) 1.756 N 0.85 1.45 0.6 645 0.716 -0.981 5. Cart (495g) + 4(50g) 1.676 N 1 1.55 0.55 695 0.602 -0.775 6. Cart (495g) + 5(50g) 2.002 N 0.85 1.45 0.6 745 0.733 -0.902 7. Cart (495g) + 6(50g) 1.768 N 0.7 1.35 0.65 795 0.633 -0.822

8. Cart (495g) + 4(50g) LIGHT 0.899 1.3 1.9 0.6 695 0.285 -0.466 9. Cart (495) + 4(50g) MEDIUM 0.833 1.4 2 0.6 695 0.427 -0.307 10. Cart (495) + 4(50g) HARD 2.559 0.75 1.4 0.65 695 1.114 -1.276 Mass Errors 0.00495 0.005472888 0.005991869 0.00650788 0.007021574 0.007533426 0.008043786 0.005991869 Velocity error 0.057662289 0.057662289 0.057662289 0.057662289 0.057662289 0.057662289 0.057662289 0.057662289 Force error 0.507894663 0.507894663 0.507894663 0.507894663 0.507894663 0.507894663 0.507894663 0.507894663 Momentum error 0.02958712 0.032607022 0.035264922 0.038797231 0.041225148 0.044689337 0.047311962 0.040327137 Impulse error 0.262764044 0.287587018 0.310722531 0.315236741 0.289762057 0.318317228 0.339979102 0.307523524