PHYSICS: TRACKER AND RC CAR LAB REPORT. Crista Falk

Transcription

1 PHYSICS: TRACKER AND RC CAR LAB REPORT One Dimensional Kinematic Analysis of a Radio Controlled Car Crista Falk Lab Partners: Jackson Bronsell, Hyrum Catanzaro, Tyler Evans AP Physics 1 B7 I. ABSTRACT The one dimensional motion of an RC car was recorded using a camera, capturing the magnitude of the car s acceleration when its controls are pressed down full-throttle. Using the Tracker software for kinematic modeling, the motion of the car could be represented graphically in terms of position, velocity, and acceleration over time. Through analysis of the digitally produced graphs, the relationship between an object s position vs. time, velocity vs. time, and acceleration vs. time during motion was demonstrated. The slope of a position vs. time graph was discovered to be equal to an object s average velocity while, similarly, the slope of a position vs. time graph during a given interval of time is equal to that object s acceleration during the same interval. Constant acceleration, as conveyed through this experiment, is represented through a curved position over time slope, a linear velocity over time slope, and a horizontal line, no slope, for acceleration over time. When represented as a function of time, acceleration on the y-axis remained at ~3, showing that the RC car underwent an average, constant acceleration of ~3 over the course of its motion.

2 II. PURPOSE The objective of this experiment is to determine the acceleration of an RC car. Furthermore, the resulting data should demonstrate how graphs of position, velocity, and acceleration as functions of time relate to each other when used to represent the same motion. III. MATERIALS 1 RC Car and Remote Control Camera (phone or other recording device) Tracker Software (available online, open-source for all common platforms) Meter stick (for purpose of calibration in video analysis) IV. PROCEDURE 1. Collect group materials, and move to a professor-approved location (so as to avoid lab interference or damage to experiment materials). 2. At location, practice driving the RC car at full throttle for 1-2 seconds, in a straight path, until the driver is confident in executing this task successfully. 3. Set up your calibration stick in view of the camera, with the RC Car beginning at the 0 mark of the meter stick. The video captured by the camera should show 4-6 meters if possible, filmed horizontally and in side view of the vehicle. 4. Keep the camera completely still while filming, carefully avoiding movement. 5. As practiced in step 3, hold down the trigger of the RC Car and film the car s acceleration. Do not end the video until the car has exited the camera s view. Repeat this process several times until a video of acceptable quality is produced. 6. Digitally transfer the chosen.mov file to a location where it may be made accessible on a personal computer, e.g. flash drive, Google Drive, cloud hosting service, etc. 7. Having downloaded the open-source Tracker software, open the video using this program. 8. Using the program tools, set the calibration stick to the meter stick from the video. Ensure that the units used in Tracker reflect the real-world quantities, i.e. one meter is properly represented in relation to the meter stick. 9. Select a frame of reference on the moving object, such as the front wheel or hood, and plot the movement of this point consistently and accurately in each frame. 10. Using the graphing function available through Tracker, use these points to represent the car s one 1

3 dimensional motion in three forms: position vs. time in the x-direction, velocity vs. time in the x-direction, and acceleration vs time in the x-direction. V. DATA & OBSERVATIONS TIME (seconds) POSITION (meters) Traveling right relative to origin (set m before object s xᵢ ) VELOCITY (m/s) ACCELERATION () s m 0 m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s s m m/s

4 0.433 s m m/s s m m/s s m m/s s m m/s s m A general trend in the data can be observed in that as the variable time increases, so too does the velocity. The object s acceleration at different points in time fluctuates around the values 2 and 3. VI. GRAPHS Fig. 1 Change in Position (m) of RC Car Relative to Time of Motion (s): Representation of constant acceleration from rest demonstrated through curved slope of position as a function of time. The upward curve of pos x over time t represents the object s displacement to the right of the origin throughout the course of its 3

5 motion. Fig. 2 Increase in Magnitude of Velocity (m/s) as a Function of Time (s): A linear slope, representing increase in magnitude of the subject s velocity over time, i.e. constant acceleration. The upward slope of function V(t) represents an increased magnitude of velocity in the positive direction. Fig. 3 Constant Acceleration () of Car at All Points During Motion (s): No change in slope of average acceleration as a function over time, indicating constant acceleration at a magnitude of ~2.8 m/s ² 4

6 to the right of the origin. VII. CALCULATIONS Fig. 1. Position Vs. Time: V av Δd / Δt m m s 3.32 m/s V inst of point (0.2 s, m) slope of tangent line ~ 2 m rise 0.6 s run 3 m/s Fig. 2. Velocity Vs. Time: V ᵢ+V f m/s 2 V av 2.06 m/s ~ 2 m/s a av Δ v / Δ t 4.12 m /s 2.8 m/s 0.5 s 2.64 m/s ² ~ 3 m/s ² b₁+b₂ 2 4 m/s m/s 2 Δx h (0.5 s) m ~ 2 m Fig. 2. Velocity Vs. Time: aᵢ+af a av m/s ² ~ 3 m/s ² VIII. ANALYSIS QUESTIONS Position Vs. Time ( fig.1) Because the slope of a position vs. time graph reflects the object s velocity, the slope of fig. 1 can be used to determine the RC car s average velocity. The graph s upwardly curved slope reveals a linear increase in the object s positive velocity throughout its motion. The change in position, or the difference between final and initial position, over time can be used to calculate the RC car s average velocity: 3.32 m/s. Similarly, the instantaneous velocity of a single point in time can be derived from slope of that point s tangent line on a position vs. time graph. Using fig. 1, the instantaneous velocity at 0.2 seconds into the object s motion can be said to equal approximately 3 m/s. Velocity Vs. Time (fig. 2) In reference to fig.2, the slope of a velocity vs. time graph can be used to extrapolate an object s velocity 5

7 over time, i.e. acceleration. Furthermore, the area between the function line and the x-axis can be calculated to determine the object s displacement after it completes its motion. Using points on fig. 2, the RC Car s average velocity can be found by calculating the sum of the final and initial velocity and dividing by two. In this case, the average velocity of the RC car s motion can be said to equal roughly 2 m/s. Geometrically, the displacement of the RC car from start to finish can be found to equal about 2 m. Individual points on a velocity vs. time graph convey instantaneous velocity throughout an object s motion. Comparing fig. 2 s more precise representation of instantaneous velocity at m/s, 0.2 seconds into the RC car s motion, the approximation of ~3 m/s made previously using fig. 1 can be confirmed as valid. The average acceleration of the RC car throughout its motion can be found by finding the slope of fig. 2, the change in velocity over time. Through these calculations, the RC car can be said to have accelerated at an average rate of about 3. Acceleration Vs. Time (fig. 3) Through analysis of the RC car s acceleration over time graph, conclusions about the object s trend in acceleration over time. Visually, the acceleration appears to fluctuate slightly from point to point, yet the overall function of a(t) lingers approximately around the value of 3. A roughly horizontal slope on an acceleration vs. time graph reveals the object s acceleration to be constant. The variances in instantaneous acceleration can be attributed to a number of uncontrolled variables in the experiment: friction (both in terms of air resistance and traction between the tires and ground limiting the extent of the object s acceleration), human inaccuracy in using the vehicle s remote control trigger to fully and consistently accelerate the car, to a mere inability of the computer program to perfectly map the object s motion given the image and frame-by-frame position input it was given. IX. CONCLUSION By filming the one dimensional motion of an RC car accelerating for about two meters and analyzing the video with digital motion tracking software, the motion of an RC car was able to be quantified and graphed. Through this experiment, the acceleration of an RC car can be concluded to equal approximately 3. Furthermore, the relationship between one dimensional motion graphs for an object in constant acceleration could be determined. In the event of constant acceleration, such as in this experiment, the slope of the position vs. time graph will appear curved while the velocity vs. time graph will be linear and the acceleration vs. time graph should follow a horizontal trend. These conclusions have real world application in that graphs representing similar, constant acceleration in one dimension should reflect these relationships. The initial trial of this experiment only reflected the RC car s velocity mid-motion and thus had to be reconducted to produce meaningful data. Subsequently, the procedure was modified to emphasize filming from a side view and ensure that the recording of the object s motion captured its initial rest in order to reflect an accurate acceleration over time. The calculated acceleration of 3 m/s 6

8 makes sense in the scope of reality. The duration of the RC car s motion occurred for just over half a second, and the displacement of the RC was said to equal 1.6 m (approximately 2 m when considering significant figures). Reasonably, one could infer that, continuing its current trajectory, the object would end up achieving 3 meters in about 1 second. Regardless, there were still sources of error that could have arisen in the form of imprecision when tracking the motion of the object using video motion tracking software and imperfections in steering the RC car in a straight line as well as recording the motion itself. This lab has taught that motion graphs can be used to reflect different data about a moving object, but from their points and slopes information pertaining to direction, displacement, velocity (both instantaneous and average), and acceleration can be derived. If I were to repeat this lab in the future, I would improve this procedure by filming a larger range of motion (perhaps 10 meters rather than 2) and filming with a tripod to ensure accuracy of the recorded motion. 7

9 PHYSICS: Cart and Ramp Lab Report Two Dimensional Acceleration of a Cart Down a Ramp Crista Falk Lab Partners: Jackson Bronsell, Hyrum Catanzaro AP Physics 1 B7 I. ABSTRACT Through the use of a Vernier track, motion detector, and LabQuest 2 data analysis device, the average horizontal acceleration of a cart was calculated. With repetitions of the experiment at a ramp angle of 15º as well as a 36º ramp, both with and without a 0.5 kg mass attachment, the variables that affect the x-component acceleration of an object down a ramp could be determined. At an angle of 16º, the cart without a mass attachment displayed an average acceleration of while its acceleration with the mass came out to be At 36º the acceleration with and without an attached mass came out to be and respectively. Based on these values, it can be supposed that increasing the angle of the ramp rather than changing the mass of an object traveling down the ramp will increase the object s acceleration down the ramp. Multiplying the average horizontal acceleration of an object down a ramp with the sine of that ramp s angle, the vector component corresponding with the acceleration of gravity can be derived. Because the calculated results of gravitational acceleration come out to be , , , and , which are relatively close to the accepted value of -9.8, gravity is assumed to be constant. Therefore, the speed of an object down a given ramp (excluding air resistance and other frictional forces) can be said to vary primarily upon the angle of that ramp. 8

10 II. PURPOSE This lab should allow the scientist to calculate gravity s effect on an object which travels down a ramp and, furthermore, how such an object s motion differs in relationship to the angle of the ramp it travels down. For instance, this lab should demonstrate why a ramp may appear to lessen the impact of gravitational force on an object s motion. Furthermore, this lab should convey the significance of using diagrams to understand physical phenomena and the use of vector components in calculating complex (two-dimensional) motion problems. III. MATERIALS Ramp (Vernier Track) Vernier Cart Vernier Motion Detector Leveling Device (to find angle of ramp) Ring Stand Bracket Motion Detector Bracket LabQuest 2.5 kg mass IV. PROCEDURE 1. Having organized the proper lab materials, set up the ramp, positioning the ramp upright by attaching the Vernier rod clamp to the ring stand rod. 2. Attach the motion detector to the top of the track with the motion detector bracket, checking to ensure that the device is aimed straight down the track rather than too high or low. The LabQuest 2 electronic device should be plugged in to the motion detector and switched on. Set the Cart Setting to increased accuracy, depicted by the cart icon rather than the ball icon. 3. Using a leveling device, either physically or with the use of an electronic app, adjust your track until it is at an angle between 10º and 20º. 4. Place your cart at the top of the track, approximately 20 cm away from the front of the motion detector. 5. Release the cart, allowing it the travel down the ramp. As this happens, press the play button on the LabQuest 2 device to begin tracking its motion. 6. Catch the cart once it reaches the bottom of the ramp and quickly stop your motion detector once the cart stops. 9

11 7. The image displayed on the screen of the LabQuest devices should resemble the following when graphed as a function of time in the horizontal direction and acceleration in the vertical direction: 8. Highlight the section of the graph where the cart is accelerating and select the Analyze option in the top directory, followed by Statistics and Acceleration. This should allow you to find the average acceleration during that time interval, referred to as the mean, which will be located on the right side of the screen under Acceleration. 9. Repeat steps four through eight until a total of 10 trials is reached, disregarding and retesting any apparent outliers in the recorded data. 10. Taking not of the current angle of the ramp, record the results of the first experiment in a data table, and calculate the average acceleration of the ten trials. 11. After attaching the.5 kg mass to your cart, repeat steps four through ten at the same angle as the previous experiment. 12. After the first two charts have been filled out, readjust the ramp to an angle between 25º and 40º then repeat the previous processes, first without the.5 kg mass and then with. Record the results of ten trials each and then calculate the average acceleration for both. 13. Based on mathematical knowledge of trigonometric relationships, calculate the acceleration of gravity.* * Note that the values calculated in the experiments are the x-components of gravity for each different angle, which act as adjacent acceleration components in the horizontal direction (which runs parallel to the ramp). The value of acceleration of gravity (a g ) can be determined through right triangle relationships as long as the values of one side adjecent hypotenuse and one angle are known. Therefore, by plugging in the average x-component of gravity into a sin θ ax ag equation, in which the reference angle θ is the angle of incline of the ramp, one can solve for the component of acceleration due to gravity. 10

12 V. DATA & OBSERVATIONS Experiment #1 - Acceleration () of Cart Ramp Angle: 15 Trial Cart Cart +.5 kg Average Acceleration Without Mass: Average Acceleration With Mass: Experiment #2 - Acceleration () of Cart Ramp Angle: 36 Trial Cart Cart +.5 kg Average Acceleration Without Mass: Average Acceleration With Mass: VI. GRAPH a g represents the force of gravity acting upon an object, which is theoretically said to equal a g-x represents the horizontal component of an object s acceleration as a result of gravity, moving parallel to the slope. a g-y represents the vertical component of an object s acceleration, perpendicular to the slope and the horizontal vector component, a g-x. 11

13 VII. CALCULATIONS opp sin θ hyp sin θ Average Acceleration in x direction Acceleration of Gravity The angle of the ramp is opposite of the x-component of gravity. The average x-component of gravity can be plugged into the sin equation. θ ( A cceleration of Gravity ) sin Average Acceleration in x-direction Average Acceleration in x direction A cceleration of Gravity sinθ Gravity for Experiment #1 - Without Mass: sin Acceleration of Gravity a g Gravity for Experiment #1 - WithMass: sin Acceleration of Gravity a g Gravity for Experiment #2 - Without Mass: sin Acceleration of Gravity a g Gravity for Experiment #2 - Without Mass: sin Acceleration of Gravity a g VIII. ANALYSIS QUESTIONS In physics, two dimensional kinematics are calculated by resolving an object s motion into components or individual vectors based on the axes on a Cartesian coordinate grid. Two-dimensional vector components are commonly referred to as the horizontal and vertical components, much like in this experiment. Through the use of diagrams, phenomena in physics can be represented conceptually to allow for a deeper understanding of how parts of motion interact with each other (for example the interaction of horizontal, gravitational, and vertical components of an object traveling down a ramp). Although an object going down a ramp is under the influence of gravitational force, it travels at an angle in a horizontal as well as vertical direction and therefore does not undergo the full acceleration of -9.8 that an object in vertical free fall will. As demonstrated by this experiment, the acceleration due to gravity remains constant as a vector component. After performing the above computations, the acceleration of gravity is found to equal roughly -10. Moreover, the acceleration of an object moving down a ramp 12

14 can be said not to depend upon the mass of the object as evidenced by the similarities of the calculated acceleration between a regular cart moving down a ramp of 15 and a +.5 kg cart moving down a ramp of 15 (roughly 2.5 ). However, the horizontal acceleration value of each of the 36 angle ramp tests comes out to be approximately 5.6. Thus, one could conclude that changing the angle of a ramp affects an object s rate of acceleration, not the force of gravity upon the object (which is constant) nor the object s mass (which is negligible in conditions that ignore frictional forces such as air-resistance). The acceleration of an object down a steeper ramp will have a greater magnitude than an object s acceleration when traveling down a less steep ramp. Therefore, it can be concluded that as the angle of a ramp increases to approach 90, the object s acceleration down the ramp will likewise increase to approach the value of gravity s acceleration (since a perfectly vertical ramp will essentially become freefall). IX. CONCLUSION Using LabQuest and motion detecting tools, the acceleration of a Vernier cart down an angled track was recorded and analyzed. By resolving the motion of a cart traveling down a ramp into individual horizontal and gravitational vector components then performing trigonometric calculations (dividing the cart s horizontal acceleration by the sine of the ramp s angle), the gravitational acceleration component of the cart s motion could be derived. Because the resultant value of , , , and are similar to the accepted value of gravity s acceleration, ~ -10, the outcomes of this experiment can be logically affirmed. One could conclude from the results of this experiment than the cart accelerated due to the action of gravity but was constrained to accelerate parallel to the surface of the ramp. Because this factor varies often in slope while gravity is roughly constant around the world, we know the ramp of the angle down which an object travels affects the object s acceleration. However, potential sources of error in this lab may have arisen due to imperfections in the equipment, such as slight friction between the cart s tires and the track, air resistance, and human inconsistency in choosing the interval of points from the LabQuest graph with which to calculate average acceleration. With this in mind, future repetitions of this experiment should be conducting which utilize a more consistent method of starting and stopping the cart, such as moving a block out from the front of the cart rather than allowing it to fall from your hands. A more systematic method of starting and stopping the motion analysis device along with calculating acceleration using a definite number of points could be worth implementing as well. Applying these improvements might reduce any unaccounted variables to the cart s motion and allow for more credible calculations. 13

15 PHYSICS: PROJECTILE MOTION LAB REPORT Analyzing motion components and trajectory of a projectile launched at an angle Crista Falk Lab Partners: Jackson Bronsell, Hyrum Catanzaro, Tyler Evans AP Physics 1 B7 I. ABSTRACT In this experiment, a projectile launcher and LabQuest 2 interface were used to explore accurate methods of calculating range of a projectile, specifically when launched from above ground. A steel ball was shot in repeated trials at various angles to demonstrate the effect of launch angle and exit velocity on the horizontal distance a projectile will travel. A few major conclusions were drawn about the nature of projectile motion: 1.) The range equation is only valid for determining the motion of a projectile that starts and stops on the same plane. Otherwise, a derivation of the kinematic formulas is necessary. 2.) The optimal angle for achieving the farthest range is ) The kinematic formulas can be used to predict the range of a projectile given an exit velocity and launch angle and resolving the motion into vertical and horizontal components. 14

16 II. PURPOSE This experiment should effectively demonstrate calculated measures involved in projectile motion problems such as initial velocity, time, range, and initial angle. Throughout the experiment, data will be collected as it pertains to trajectory of a projectile at various angles such as range and velocity. Discrepancies between theoretical values and experimental values will be calculated and explained through logic and evidence. At the conclusion of this experiment, the student should be prepared to launch a steel ball into a plastic dipping sauce cup in a single try given a random angle between 10 and 80. The pressure and range of the cup are to be determined by the group, and the process or equation used to estimate the projectile s horizontal displacement may be derived from the kinematic formulas. III. MATERIALS LabQuest 2 Interface Projectile Launcher Set Launcher Air Pump Steel Spheres Marking Tape - (about 30 cm) Stopwatch (cell phone or LabQuest 2) Projectile Stop Meter Sticks and/or Metric Tape Measures IV. PROCEDURE 1. Set Up a. Attach the air pump to the launcher. b. Set up the launcher to the desired angle. c. Attach the cable to the launcher and then attach to the DIG 1 or DIG 2 slot on the LabQuest 2. d. Insert the steel ball. e. Pump up the air pressure and turn on the LabQuest (The group should decide on a constant PSI, such as 100 psi). f. To Fire, hold arm and then press launch. 2. Part 1 - Angles and Ranges 15

17 a. Test the range of your Projectile Launcher as it relates to the angle at which it is launched. b. Place your launcher on the ground. c. While using a tape measure and the marking tape, find the range of the projectile for each of the angles you test. (You may test an angle more than once.) d. Maintain the same pressure throughout (such as 100 psi), and do a total of 10 test shots. 3. Part 2 - Calculating Range a. Using the range equation (see section VII.), find the calculated range of the projectile as it leaves the launcher based on the exit Velocity (Vₒ) given to you by LabQuest 2. i. Press play on the LabQuest 2 in order for it to give you Vₒ as the projectile ii. iii. launches. Have a lab partner measure the actual (experimental) range for each trial. You may choose to keep the PSI and angle consistent for each launch or you may change them. Calculate the estimated range for each shot. 4. Part 3 - Finding Exit Velocity Without a Timer a. Place your launcher on the lab table and make sure that the launcher is aimed so that it will fire the objects horizontally. 1 b. Record the launch height and use the Δ x v 0 t + 2 at ² to find the time of fall for any object (neglecting air resistance) falling from the set height with a 9.8 m/s ² for the acceleration due to gravity. c. Once you have the time of fall, run a series of tests to compare the calculated exit velocity of the projectile to the actual exit velocity as given by LabQuest 2. d. For each launch, press play on the LabQuest 2 so it can determine the exit velocity of the steel ball as it leaves the chamber. e. Use fall time and the actual range (measured by using marking tape and a tape measure) to calculate exit velocity based on experimentation. 5. Part 4 - Launching into a Plastic Cup a. Using the information on the back of the Projectile Launcher (height off the ground, for example), come up with an equation or process that will allow you to find the estimated range of a projectile that is launched from a specific height above ground level. b. Find a consistent exit velocity of your projectile launcher. Your group may choose the PSI and therefore exit velocity. You may also place the plastic cup where you think the ball will land after doing the necessary calculations. Given a random angle between 10 and 80, you will need to land a steel ball into a plastic cup in one try. 16

18 V. DATA & OBSERVATIONS Part 1 - Angles and Ranges Test Angle Range (m) m m m m m m m m m m Angle that produced the farthest range: 45 Angle that produced the shortest range: 0 Angle that produced the greatest height: 90 17

19 Part 2 - Calculating Range Test Θ Vₒ Estimated Range Actual Range m/s m m m/s m m m/s m m m/s m m m/s m m m/s m m m/s m m m/s m m m/s m m m/s m m Part 3 - Finding Exit Velocity Without a Timer Test Actual Range Vₒ - from LabQuest Calculated Vₒ m m/s m/s m m/s m/s m m/s m/s m m/s m/s m m/s m/s m m/s m/s m m/s m/s m m/s m/s m m/s m/s m m/s m/s 18

20 VII. CALCULATIONS Part 2 - Calculating Range: Range 2 νₒ * sin2θ g Equation Result Residual (e R - R) 4.836² * sin2(0) ² * sin2(5) ² * sin2(10) ² * sin2(15) ² * sin2(30) ² * sin2(45) ² * sin2(50) ² * sin2(60) ² * sin2(65) ² * sin2(70) m 0 m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m m Σe Average Residual between Experimental and Actual Range: R n m m Part 3 - Finding Exit Velocity Without a Timer: 19

21 1 Δ y vᵢt + 2 at² Launch Height: m Fall Time: 0.35 s * t - 4.9t ² t ² t 0.35 Equation Δx v ᵢ t Result Residual (e V - V) v ᵢ v ᵢ v ᵢ v ᵢ v ᵢ v ᵢ v ᵢ v ᵢ v ᵢ v ᵢ m/s m/s m/s m/s m/s m/s m/s 0.39 m m/s m/s m/s m/s m/s m/s m/s 0.69 m/s m/s m/s m m/s m/s m/s m/s 0.05 m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s m/s Σe Average Residual between Experimental and Actual Velocity: V n m/s m/s Part 4 - Launching into a Plastic Cup: 20

22 Angle Given: 67 Average velocity at 100 psi: 5.62 m/s 1. Resolving Vectors: c os(θ) V ox V o c os(θ) * 5.62 V ₒₓ s in(θ) V oy V o s in(θ) * 5.62 V ₒy 2. Vertical Component: Δ y V ₒt 1 2 at² Δ y sin(θ) * V ₒt 4.9t² s in(θ) * 5.62t 4.9t² 0 4.9t² + s in(θ) * 5.62t sin(θ) * 5.62± sin(θ) * 5.62t² 4* 4.9* * t a Horizontal Component: Δx v ᵢ t t * cos( θ )*5.62 Δ x * The cup should be placed 3.7 meters from the launch origin For θ 67, t s VIII. ANALYSIS QUESTIONS 21

23 Part 1 - Angles and Ranges From this portion of the experiment, it could be determined that angles closer to 0 have a greater horizontal velocity component but a smaller vertical velocity component. Likewise, angles closer to 90 have a greater vertical velocity component and smaller horizontal component. For this reason, any projectile launched at angle θ should travel the same horizontal distance as a projectile launched at 90 - θ, assuming negligible air resistance. The magnitude of V x and V y will be reversed for 90 - θ and θ, resulting in the same V i when combined. Given that this is the case for velocity components, a projectile launched at 45, the optimal angle, will travel the farthest as it has an equal horizontal and vertical component. Part 2 - Calculating Range My estimated and actual ranges were close in value, but it was apparent that the estimated ranges were consistently lower than the experimental ranges, excluding two outliers in which the expected ranges were higher than those which were recorded by a mere m (60 ) and m (65 ). On average, the estimated ranges calculated using the range formula were m less than the distances the projectiles really launched. This repeated discrepancy can be explained in that the range equation assumes that a projectile will be launched from the same height that it lands at. However, the projectile launcher is fixed m off the ground. For this reason, the range equation will give the range of a projectile up to m before it hits the ground. Fig 2. The difference between calculated range and experimental range of a projectile when launched from a height above the ground. Part 3 - Finding Exit Velocity Without a Timer As shown by the calculations in Part 3 of Section VII, the magnitude of experimental velocity and calculated velocity are consistently close in value by less than a tenth of a m/s difference. In fact, the calculated residual of the data set is a mere m/s difference between the recorded and calculated velocities. Compared to the theoretical and experimental ranges, for example, the calculated predictions 22

24 are much more accurate. Unlike the range equation, the values used in the displacement kinematic equation (such as time and measured range) are indicative of the object s full trajectory. Any minimal discrepancies can then be explained by air resistance and real world factors ignored in the equation for ease of calculation. Part 4 - Launching into a Plastic Cup After calculating that a steel ball launched at 67, provided that we launched it at 5.62 m/s (100 psi), would have a horizontal range of around 3.7 m, the plastic cup was set up and the projectile launcher prepared. The test was successful, and the ball did land in the cup. IX. CONCLUSION Ultimately, the purpose of the lab was to gain enough insight about projectile motion to figure out the proper metrics to launch a steel ball into a cup given a particular launch angle. Being able to solve this real-world challenge provided a valuable reference for how to solve two-dimensional projectile problems that will later be applied future exams, science, and mathematics courses. A projectile launcher was used first to determine the effect that launch angle has on range. It was determined that an angle of 0 produced the shortest range, 90 produced the greatest height, and 45 produced the farthest range. It was also shown that both 90 - θ and θ will produce the same range. Then, repeated trials were conducted to determine range using the range formula compared to the actual values, demonstrating that the range formula is not ideal for trajectories that start and end at different heights. After this, the fall time of a projectile at a given height was determined using the vertical displacement kinematic formula,, which came out to be 0.35 seconds at a height of m. The value of time could then be plugged into the horizontal, constant velocity equation to compare experimental and calculated velocity of a steel ball projectile, exemplifying that the kinematic equations are better in a scenario when a projectile starts or ends at a height above ground. With this knowledge in mind, a similar method could be applied to the challenge of shooting a projectile into a plastic cup when provided with only a set launch angle. One important insight gained from this lab was that calculating horizontal motion of a projectile is inherently different from calculating vertical motion: vertical velocity experiences constant acceleration due to gravity of -9.8 whereas horizontal velocity is constant based on its component initial velocity. The results of this experiment were reasonable with no apparent outliers that would suggest significant error in executing the lab procedure. However, future iterations of this lab may be made much more accurate if more then 10 test launches could be used when collecting data about the range and exit velocity of the steel ball (part 1, 2, and 3). The more data acquired, the more ideal the calculation results would be. 23

25 PHYSICS: NEWTON S LAWS LAB Exploring Newton s Second and Third Laws Through Carts and Motion Detectors Crista Falk Lab Partners: Jordan Lo and Michela Rubagotti AP Physics S2 B7 I. ABSTRACT This report explains the interaction of carts as well as the properties of dynamics that are demonstrated when constant force is applied in different situations. The three test situations of interest are 1.) theoretical vs. observed acceleration of a cart being pulled forward by the rope of a pulley system with different hanging masses (therefore forces) attached, 2.) how equal and opposite reactions look when applied to two objects of different masses comparing speed and direction, 3.) how acceleration varies when an equal force is applied to systems of different mass. Respectively, this lab report should introduce and explain ideas of Newton s second law, Newton s third law, and momentum and impulse. 24

26 II. PURPOSE This lab is concerned with exploring the real world applications of physical dynamics, explaining rather than merely describing motion. Specifically, this lab observes the ways in which the experimental values of forces differ from their theoretical values in the context of Newton s second and third laws. This lab will also begin to introduce the concepts of momentum and impulse. Using gravity as an application of constant force for much of the experiment eliminates some of the difficulty in observing Newton s laws in a lab setting. III. MATERIALS 1 Vernier Dynamics Cart 1 Vernier Dynamics Cart with a Plunger kg gram mass with bolt and fastener 1 Vernier Pulley 1 pulley bracket 1.25 meters of string 1 set of aluminum hanging masses (box of 6 masses) 1 LabQuest 2 1 Vernier Motion Detector 1 Motion Detector Bracket 1 triple balance beam balance IV. PROCEDURE Part 1- Newton s Second Law: 1. Set up the track so that the pulley is secured with the bracket on one side, close to the edge of the lab table, and the motion detector is secured with the motion detector bracket on the other side. 2. Set the motion detector to the cart setting. 3. Set the track so that the pulley will allow for a hanging mass to fall to the ground without interference from the side of the lab table, and do not move the position of the track throughout the lab. 4. Use a leveling app or tool to calibrate your track to Measure and record the mass of your Vernier Dynamics Cart. 6. Perform calculations to determine the force of pull on your cart for each hanging mass, which is 25

27 equal to the mass of the hanging mass times acceleration due to gravity (approximately 9.8 ). 7. Then calculate the theoretical acceleration of the cart by taking the force of pull on the cart divided by the mass of the cart for each of the six hanging masses. 8. Place the cart on the track in view of the motion detector. 9. Tie the string to the cart and tie a loop at the other end. The aluminum masses will be hanged from this loop over the pulley. 10. Using the motion detector and LabQuest2, measure the actual acceleration of the cart by taking the average of three trials for each hanging mass. Graph the acceleration vs. mass for both the theoretical values and the actual values, resulting in two lines of six points each. Part 2- Newton s Third Law: 1. Remove the pulley and bracket, putting them away, and turn off the motion detector for now. 2. Place two carts on the track. Lightly roll them toward each other so that when the two carts come close to each other, the magnets cause them to repel. Observe what happens. 3. Now place one cart on the tack at rest. Lightly roll the other cart toward the stationary cart so that the magnetic fields repel each other. Try and roll the cart at the same speed as you did in the previous step. Observe whether the stationary cart appears to move faster this time than in the previous step or at the same velocity. 4. Add a kg mass to one of the carts. Once the mass id added, roll the carts toward each other and observe how they interact. 5. Place the cart with the mass on the track and leave it at rest. Roll the other cart toward it. Observe what happens. 6. Place the cart without the mass on the track and leave it at rest. Roll the cart with the masses on it toward the stationary cart and observe how it compares with the two previous steps. Part 3- Momentum and Impact: 1. Setting up the cart and track, place the plunger cart against the back stop. Remove the kg mass if it is still attached. 2. Using the motion detector, find the velocity of the cart after it is sent across the level ramp with the force of the plunger. Do this three times and record the data. a. Repeat this step for a single cart, two carts linked together, and a cart with a weight of kg attached. 3. Find the average of each set of three velocities and compare data for each cart system. V. DATA & OBSERVATIONS 26

28 Part 1- Newton s Second Law HANGING MASS FORCE OF HANGING MASS (mass times gravity) MASS OF CART PLUS HANGING MASS THEORETICAL ACCELERATION OF CART kg N kg kg N kg kg N kg kg N kg kg N kg kg N kg 1.56 MASS ACCELERATION OF CART (three trials for each mass) AVERAGE ACCELERATION kg kg kg kg kg kg

29 Part 3- Momentum and Impulse MASS OF CART VELOCITY OF CART (three attempts) AVERAGE VELOCITY kg m/s m/s m/s m/s MASS OF 2 CARTS VELOCITY OF CART SYSTEM (three attempts) AVERAGE VELOCITY kg m/s m/s m/s m/s MASS OF CART W/ MASS VELOCITY OF CART (three attempts) AVERAGE VELOCITY kg m/s m/s m/s m/s 28

30 VI. GRAPHS Fig. 1 Graph of the Observed and Theoretical Accelerations of the Cart at Different Masses : Although it would appear that the theoretical acceleration of the cart would continue in a linear trend in which acceleration is F directly proportional to the force of gravity acting upon an object (as per Newton s second law: a m ) which in this case increases as the mass does (since F weight m*g), this is not the case for the observed values of acceleration. However, the general trend is still a positive slope for mass vs. acceleration. 29

31 VII. CALCULATIONS Part 1. Newton s Second Law: Method for Calculating Force of Hanging Mass: kg Mass of Hanging Mass * 9.8 Acceleration due to Gravity F orce of P ull on Cart Newtons kg * N kg * N kg * N kg * N kg * N kg * N Method for Calculating Theoretical Acceleration of Cart: N g N g N g N g N g N g 30

32 VIII. ANALYSIS QUESTIONS Part 1. Newton s Second Law: Comparing on the calculated values of acceleration with the actual values as recorded by the motion detector, it is observed that average, real world acceleration is consistently lower. For example, for a 0.10 kg mass or force of 0.98 N, the predicted value of acceleration is found to be although the cart s observed acceleration was only Referring to Fig. 1, this trend continues. Furthermore, as evidenced by the 1.56 theoretical acceleration and 1.08 observed acceleration for a kg weight and resultant force of N. Based on these results, an apparent relationship may exist in which the greater the force applied, the bigger the discrepancy between theoretical and actual acceleration becomes. Again referring to Fig. 1, the graph of theoretical acceleration vs. mass appears to be fairly linear. However, the actual acceleration vs. mass graph deviates from this trend. In addition to being significantly lower and even roughly logarithmic in form. Two factors that may contribute to the difference between actual and theoretical acceleration for the same forces may be 1.) Friction forces acting on the system, such as kinetic friction between the wheels and ramp as well as the rope and pulley, and 2.) Air resistance, namely in the following of drag as the masses are dropped to operate the pulley (and of course the air resistance acting on the cart itself). Part 2. Newton s Third Law: When two carts are lightly rolled toward each other with the magnets facing each other such that they would repel upon impact, it is found that they will go toward each other without ever colliding, slow to a stop, change direction, and accelerate with equal magnitude in opposite directions. On the other hand, when one cart is lightly rolled toward a stationary cart with the same speed, the cart that gets hit absorbs the force of the other moving cart. The moving cart stops and the previously stationary cart moves away at the same velocity as the cart in the previous trial. For these two examples, each cart s reaction can be explained through physics. The cart with the mass attached has a greater inertia, that is, resistivity to change its motion. Although the action-reaction pair of forces (F cart1oncart2 and F cart2oncart1 ) are equal according to Newton s third law,the acceleration of the heavier cart is less than that of the lighter cart because acceleration is inversely proportional to mass when force is constant (a F/m). When the cart with the mass is left at rest with the other cart rolled toward it, the force of the first cart on the stationary cart is not enough to push the massed cart. Instead the massed cart remains at rest and the restoring force of the heavy cart back on the lighter cart pushes back, causing it to accelerate back away. 31

33 Finally, when the cart without without a mass is left at rest and and the massed cart is rolled toward it, the mass cart comes to a stop and the lighter one moves away at a higher velocity than the massed cart rolled at before impact. Part 3. Momentum and Impulse: The velocity for three systems were tested and compared to one another in terms of their masses: a single cart of kg (0.393 m/s), cart with a mass attached of kg (0.272 m/s), and two carts of kg combined (0.227 m/s). The single cart trial resulted in a higher average velocity than the double cart trial by about m/s. The velocity of the plunger cart with a mass was lower than just the plunger cart by about m/s. Similarly, the velocity of the two cart system had a smaller average velocity than the velocity of the cart with the mass by about m/s. Because the mass of two carts is exactly double the mass of one cart, my initial prediction would be to expect a velocity for the two-cart system of exactly one half the velocity of the single cart sice it would take twice the force to achieve the same acceleration theoretically. While this was not entirely true, it seems that the greater the mass, the lower the average velocity is for an object when an equal force is applied. The least massive object weighing kg had the greatest velocity at m/s. The second most massive system of kg had the second lowest velocity, m/s. The object with the most mass (1.054 kg) had the smallest velocity at m/s. Comping a single cart to the system of two, it can be observed that when an equal force is applied to an object of two times another object s mass, its velocity would be a little over half the others (57.8% in this case) 32

34 IX. CONCLUSION At the conclusion of this lab, insights were able to be made regarding the real-world application of physical dynamics. A few general points were made: Firstly, and most obviously, there is almost always a difference between theoretical and actual values. To test this, two vernier carts, a pulley, a mass, and a motion tracker were used. In part one, a pulley with varying weights attached was attached to a single cart in order to utilize gravity as a constant force, and therefore constant acceleration. The predicted values of acceleration were determined by performing calculations using Newton s second law and student knowledge of how to calculate forces in a system. These two acceleration values, both actual and theoretical, for each hanging mass used in the pulley system were graphed, leaving the student to infer why these two values are not equal. When graphed, it was clear that these values followed a similar overall slope, but a clear discrepancy was apparent: when a 0.1 kg weight was attached, the theoretical value of acceleration was 1.56 although the observed acceleration was closer to 1.08, for example. These differences might be explained because the calculations fail to account for friction of the wheels on the cart, the friction of the rope in the pulley system, and air resistance applied to the cart as well as drag of the hanging mass pulling the pulley. Secondly, Newton s third law was explored by testing the results of colliding two carts in different situations: acceleration and velocity when two carts collide at equal velocity, when heavier cart strikes a stationary, lighter cart, when a moving light cart hits a heavier cart, et al. In all cases, the system was found to display conservation of momentum but different interactions between the carts. This could be explained because heavier objects have greater inertia, or resistivity to change their motion, so it takes more force to move than a lighter cart. However, according to laws of equal and opposite reactions, an equal and opposite force is applied both on the lighter cart from the heavier cart as is applied to the heavier cart from the lighter cart. This is why the lighter cart moves at a higher velocity while the heavier cart may not move and two carts of equal weight while travel away from each other with equal velocity. Finally, the laws of momentum and impulse were briefly demonstrated by measuring the acceleration of a single cart, two carts linked together, and a single weighted cart using a motion tracker. It was found that generally, the higher the mass, the lower the average velocity of the object will be as long as the same force is applied. 33