PHYS 103 LAB REPORT: KINEMATICS ASSIGNMENT INSTRUCTIONS OVERVIEW Laboratory reports are an essential component of a physics education. INSTRUCTIONS Download this document and record the results for each table. Activities 13 are to be performed for this exercise. All photos, data tables, and discussion must be submitted within a single 12- page Word document. Acvi 1 Submit a photo of the nal graph of data table given in Activity 1 labeled with your name and the date. Acvigy 2 Record time at various positions below in Activity 2, Table 1 per the details in Activity 2 laboratory exercise instructions. Graph the points from Activity 2 Table 1 and draw a line of best t and determine the slope of the line. Graph paper is included below or is available by printing from the laboratory exercise instructions. This is the speed of the car. Submit a photo of the graph and your work to determine the speed of the car. Data Table 1 Time (5) Displacement (m) 2.00 Speed of car After determining the velocity of the car, you may continue to Activity 3. Activity 3 Use the steel sphere to make measurements and complete Table 3. Data Table 3 PHYS 103 Time (s) _ Average lime (s) _ Average Time 2 (52) Distance (111) Trial 1 = Trial 2 = Trial 3 = Trial 1 = Trial 3 Trial 1 Trial 3 = Trial 2 = Trial 2 = 0.1 Trial 1 = Trial 2 = Trial 3 = Trial 1 = Trial 3 = Trial 2 = Trial 1 = Trial 2 = Trial 3 = Trial 1 = Trial 2 = Trial 1 = Trial 3 = Trial 3 = Trial 2 = 0.8 Submit completed table Submit photo of setup showing inclined angle and the protractor (similar to Figure 5 in the exercise). Submit graph of displacement vs time squared from Table 3. ll Preparation 1. Collect materials needed for This investigation. 2. Locale and clear an area of level floor space in order to conduct the constant velocity experiment. The space should be tree of obstruction and three to tour meters long with a surface which will allow the vehicle to maintain traction but not impede the vehicle. Activity 1: Graph and interpret motion data of a moving object One way to analyze the motion of an object is to graph the position and time data. The graph of an object's motion can be interpreted and used to predict the object's position at a future time or calculate an object's position at a previous time. Table i represents the position of a train on a track. The train can only move in one dimension, either forward {the positive x direction} or in reverse [the negative x dheconl Table 1 Time x-axis ,seconds Position -axis , meters 0 O 5 20 TO 40 T5 50 20 55 30 60 35 7O 40 7O 45 7O 5O 55 1. Plot the data from Table l on a graph using the yaxis to represent the displacement from the starting position [y = O} and the time coordinate on the xaxis. 2. Connect all the coordinates on the graph with straight lines. 12 Activity 2: Calculate the velocity of a moving object In this activity you will graph the motion of an object moving with a constant velocity. The speed of the object can be calculated by allowing the Constant Velocity Vehicle to travel a given distance and measuring the time that it took to move this distance. As seen in Activity l, this measurement will only provide the average speed. In this activity, you will collect time data at several travel distances, plot these data, and analyze the graph 1. 2. 99???" 1]. Find and clear a straight path approximately two meters long. Install the batteries and test the vehicle. Note: The vehicle should be able to travel two meters in a generally straight path. If the vehicle veers significantly to one side, you may need to allow the vehicle to travel next to a wall. The friction will affect the vehicle's speed, but the effect will be uniform for each trial. Use your tape measure or ruler to measure a track two meters long. The track should be level and smooth with no obstructions. Make sure the surface of the track provides enough fraction for the wheels to turn without slipping. Place masking tape across the track at 25 cm intervals. Set the car on the floor approximately 5 cm behind the start point of the track. Note: Starting the car a short distance before the start point allows the vehicle to reach its top speed before the time starts and prevents the short period of acceleration from affecting the data. Set the stopwatch to the timing mode and reset the time to zero. Start the car and allow the car to move along the track. Start the stopwatch when the front edge of the car crosses the start point. Stop the stopwatch when the front edge of the car crosses the first 25 cm point. Recover the car, and switch the power off. Record the time and vehicle position on the data table. . Repeat steps #59 for each 25 cm interval marked. Each trial will have a distance that is 25 cm longer than the previous trial, and the s opwatch will record the time for the car to travel the individual trial distance. Record the data in Data Table I. 13 Data Table l Time (5) Displacement (m) 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 l2. Graph the Time and displacement data points on graph paper. 13. Draw a line of best fit Through the data points. Note: The points should generally fall in a straight line. If you have access to a graphing calculator or a computer with spreadsheet software, The calculator or spreadsheet can be programmed to draw the line of best fit, or trend line. l4. Calculate the slope of the line. Note: Based on The equation of a line That cross the yaxis at y = O, the slope of the line, m, will be the velocity of the object. y = m d = wilt 15. Make a second data table, indicating the velocity of the car at any time. Data Table 2 Oa'h-IO-U'I-hWM' I4 Note: Because the object in this example. the batterypowered car, moves with a constant speed, all the values for the velocity of the car in the second table should be the same. The value of the velocity for the car should be the slope of the line in the previous graph. 16. Graph the data points from the Data Table 2 on a second sheet of graph paper. Label the yaxis Velocity and the xaxis Time. Note: When the data points from this table are plotted on the second graph, the motion of the car should generate a horizontal line. On a velocity vs. time graph, an object moving with a constant speed is represented by a horizontal line. 17. Draw a vertical line from the xaxis at the point time = 2 seconds so that it intersects the line representing the velocity of the car. 18. Draw a second vertical line from the xaxis at the point time = 4 seconds so that it intersects the line representing the velocity of the car. 19. Calculate the area represented by the rectangle enclosed by the two vertical lines you just drew, the line for the velocity of the car, and the xaxis. An example is shown as the blue shaded area in Figure 4. t1 t: t Figure 4 15 Note: In order to calculate the area of this rectangle, you must multiply the value for the time interval between time t=2 s and time 124 s, by the velocity of the car. This area represents the distance traveled by the object during this time interval. This technique is often referred to as calculating the \"area under the curve\". The graph of velocity vs. time for an object that is traveling with a constant acceleration will not be a horizontal line, but using the same method of graphing the velocity vs. time and nding the \"area under the curve\" in a given time interval can allow the distance traveled by the object to be calculated. Distance = velocity x time In this equation. the time units is] cancel out when velocity and time are multiplied, leaving the distance unit in meters. 16 Activity 3: Graph the motion of an object traveling under constant acceleration Collecting data on freetalling objects requires accurate timing instruments or access to a building with heights of several meters where objects can safely be dropped over heights large enough to allow accurate measurement with a stopwatch. To collect usable data, in this activity you will record the time objects to roll down an incline. This reduces acceleration to make it easier to record accurate data on the distance that an object moves. 1. Collect the following materials: Steel Sphere Acrylic Sphere Angle Bar Clay Tape Measure liming Device Protractor 2. Use the permanent marker and the tape measure to mark the inside of the angle bar at lcm increments. 3. Use the piece of clay and the protractor to set up the angle bar at an incline between 5" to 10". Use the clay to set the higher end of the anglebar and to stabilize the system. {Figure 5] Figure 5 17 Set up the angle bar so that the lower end terminates against a book or a wall, to stop the motion of the sphere [Figure 6.} Figure 6 4. Place the steel sphere IO cm from the lower end of the track. 5. Release the steel sphere and record the time it takes for the sphere to reach the end of the track. 6. Repeat steps #45 two more times for a total of three measurements at a starting point of 10 cm. 7. Repeat steps #46, increasing the distance between the starting point and the end of the track by 10 cm each time. 8. Record your data in Data Table 3. Note: You are recording the lime it takes for the sphere to accelerate over an increasing distance. Take three measurements for each distance, and average the lime for that distance. Record the lime for each attempt and the average time in Table 4. Data Table 3 Time (s) _ Average time (s) _ Average Time 2 (52) Distance (m) Trial I = Trial 2 = Trial 3 = 0.1 Trial I = Trial 2 = Trial 3 = 0.2 Trial I = Trial 2 = Trial 3 = 0.3 Trial T = Trial 2 = Trial 3 = Trial I = Trial 2 = Trial 3 = 0.4 Trial I = Trial 2 = Trial 3 = Trial T = Trial 2 = Trial 3 = Trial I = Trial 2 = Trial 3 = 0.8 18 9. Calculate the average time for each distance and record this value in Table 4. 10. Create a graph of distance vs. time using the data from Table 4. 11. Complete Table 4 by calculating the square of the average time for each distance. 12. Create a graph of displacement vs. time squared from the data in Table 4. l9 Graphing the displacement vs time data from Table 4 will generate a parabola. When data points generate a parabola, it means the y value is proportional to the square of the x value, or: you? That means the equation for a line that fits all the data points looks like: y=AxZ+Bx+C. In our experiment, the y-axis is displacement and the x-axis is time-: therefore displacement is proportional to the time squared: 5-0th So, we can exchange y in the equation with displacement [5}, to give a formula that looks like: s=At2+Bt+C. We would know the displacement 5', at any time t. We just need to find the constants, A, B, and C. The equation that describes the displacement of an object moving with a constant acceleration is one of the kinematics equations: 1 s = EaAtz + 13111: The following section describes how to find this equation using the same method of finding the "area under the curve" covered in Activity 2. 20 Finding an Equation for the Motion 0! an Object with Constant Acceleration The general form of a line is: y = mx + b Where m is the slope of the line, and b is the yintercept, the point where the line crosses the yaxis. Because the first data point represents time zero and displacement zero, the yintercept is zero and the equation for the line simplifies to: Y = mx The data collected in Activity 3 showed that: SOCEZ This means that the displacement for the object that rolls down an inclined plane is can be represented mathematically as: s=kt2+c Where it is an unknown constant representing the slope of the line, and c is an unknown constant representing the yintercept. The displacement of the sphere as it rolls down the incline can be calculated using this equation, if the constants kand c can be found. Further experimentation indicates that the constant ktor an object in freefall is onehalf the acceleration. If the object is released from rest, the constant c will be zero. 80 for an object that is released from rest, falling under the constant acceleration due to gravity, the displacement from the point of release is given by: 1 s = at2 2 Where sis the displacement, tis the time of freefall, and a is the acceleration. For objects in freefall near Earth's Surface the acceleration due to gravity has a value of 9.8 "1/52. Anotherway to derive this equation, and find the values for k and c, is to consider the velocity vs. time graph for an object moving with a constant acceleration. Remember the velocity vs. time graph for the object moving with constant velocity from Activity 2. If velocity is constant, the equation of that graph would be: 17 = k 21 Where v represents the velocity, plotted on the y-axis, and k is the constant value of the velocity. Plotted against time on the x-axis, this graph is a horizontal line, as depicted in Figure 7. t1 t2 Figure 7 By definition, the shaded area is the distance traveled by the object during the time interval: At = t2 - t1 displacement S time At : S = vAt If an object has a constant acceleration, then by definition: V2 - V1 a = - At Or : V2 = ant + V1 This is equation is in the general form of a line y = mx + b, with velocity on the y- axis and time on the x-axis. The graph of this equation would look like the graph in Figure 8. Velocity vs. Time Az [ m/s) A1 t2 Figure 8 Similar to how the shaded area An in Figure 7 represents the distance traveled by the object during the time interval At= to - t, the shaded area A combined withA1 equals the distance traveled by the object undergoing constant acceleration. The area A1 can be given by: A1 = \"1A: The area A2 can be given by: 1 A2 = EC\";- \"1)t Because this is the area of the triangle, where the length of the base is Atand the height of the triangle is (1?; 12'}, Adding these two expressions and rearranging: 1 s = 5072 vlt And substituting: v2 2 act + 191 Gives this equation: 1 s = i(a\\t+ vi + 132111: + tam] Simplifying gives: 1 2 3 25.11111: + 13111: This equation gives the theoretical displacement for an object undergoing a constant acceleration, a, at any time t, where sis the displacement during the time interval, at, and In is the initial velocity. lithe object is released from rest, as in our experiment, n = O and the equation simplifies to: 1 s =milt2 2 22