Physics 195: Linear Kinematics Page 1 of 6 Linear Kinematics with Constant Acceleration Objectives To understand the relationship between displacement, velocity and constant acceleration; To determine the average acceleration of an object by multiple techniques; To become familiar with various graphical representations of the state of an object in motion. Hypothesis Any object that moves in a straight line with constant acceleration will display certain characteristics that can be used to calculate the magnitude of that acceleration. Graphical representation of the data allows the acceleration to be determined in multiple ways. Overview An air-track is used to create a low friction environment. The track is inclined at a constant angle. A glider is released from rest near the top of the track and a tape timer is used to record data about the glider position at different times. In addition to numerical analysis, you will produce and analyze graphical data of the glider's motion. Equipment List Air Track, Blower & Hose, Jack, 1 & 2 Meter Sticks, 30 cm Ruler, Glider, Glider Accessory Kit, '50 gram' Masses, Ring Stand, Tape Timer, White Paper Tape, Blue Painter's Tape Mathematical Models/Reference Values Objects moving along surfaces with constant angles of inclination will display rates of velocity change (acceleration) proportional to the angle of inclination (8) and the gravitational effect of the earth (g). g has a standard value of 9.8 meters per second o ; . wp 1y % Do ; : squared. Position as a function of time: xi:x0+voxt+5 d.t' and velocity as a function of time: v, =v_ +d,t both depend on this acceleration: | |=gsin (&) . Experimental Procedure Turn on the blower and adjust the output to maximum. Center the glider at the 100 centimeter mark. If the track is level, the glider will remain motionless or slowly drift back and forth. If the glider moves towards one end of the track, adjust the two leveling screws in the support bar and repeat the process. Once the track is level, turn off the blower. Raise the track and center the jack underneath the single support leg on the air track. Expand the jack until it has a height of about 10 centimeters. 1. Measure the distance (D) between the center of the support leg and the support bar and verify that it is 100 centimeters. Measure the height (H) of the jack. Record both values in Data Table One. rev 1222 Physics 195: Linear Kinematics Page 2 of 6 2. Place the glider about 5 centimeters from the raised end of the track. Adjust the height and angle of the tape timer to be parallel to the top of the glider and about 1 centimeter above it. 3. Unroll the white paper tape until you have a piece slightly longer than the air track, then detach it from the roll. Thread the paper tape through the slots on the tape timer and beneath the carbon paper disc. Use blue tape to attach the paper tape to the top of the glider. Take the four '50 gram' masses from the glider kit and place two on each side of the glider. 4. One group member should be ready to stop the glider from hitting the end of the track. Another places their finger on the track in front of the glider to stop it from moving. Turn on the blower and wait about 10 seconds for the air pressure to reach maximum. 5. Make sure the white paper tape will move smoothly through the tape timer, then set the tape timer switch to 10 Hz. The group member blocking the glider removes their finger and the glider should slide downhill. Catch the glider as it reaches the bottom of the track, then turn off the blower. 6. Your data is a series of dots on the underside of the white paper tape. The dots are small and can sometimes be faint. Identify any missing or 'double-dotted' data points. If you can find at least 20 to 25 consecutive data points then your data is probably satisfactory. Notify the instructor. Your instructor will choose one of the data points to act as your x =0 and t = 0. 6. Attach the paper tape to a two-meter stick. Use the point selected by your instructor as x = 0 and measure the position of each other data point. Record the results in the first column of Data Table Two, then complete the next two columns. Columns five and six will be addressed later in the analysis. Data Analysis The simplest possible model for how the glider moves assumes a constant acceleration which is proportional to the angle at which the track is inclined: |d,|=gsin(#)= g% . Q1. Use the values from Data Table One to calculate the maximum possible acceleration based on this simple model. Transfer the result to Data Table Three under the heading of amooeL. The raw data collected is likely to contain random errors, but it can still give a sense of the expected results. The acceleration depends on the change in the instantaneous velocities, but you can only calculate average velocities from your data since 0.1 seconds is not 'infinitesimal'. The mean-value theorem of calculus offers a solution. For a constant slope interval, the average value of the function is the same as the instantaneous value of the function at the midpoint of the interval. So for your data, v, _=v when t=0.05 s, 0.15 s, etc. Q2. Use the velocities at t = 1.35 and t = 0.05 seconds to calculate the average ViaeV acceleration: ||z% avg ' instantaneous . Record this result in Data Table Three under agaw- Physics 195: Linear Kinematics Page 3 of 6 Graphs are the best way to look for trends in a large set of data. Before starting, read the graphing instructions in the course resources. Label downhill as the positive direction. Q3. Use Cartesian coordinates to graph the position of the glider as a function of time. On the graph, draw a smooth curve that passes through the bulk of your data - though it may not pass through each data point. Draw the line tangent to the curve at t = 0.4 seconds. Draw the line tangent to the curve at t = 1.0 seconds. Extend both tangent lines across the entire graph. Q4. The slope of these two tangent lines represents the rate of change of the glider's position - i.e. the glider's instantaneous velocity at those two times. On the graph, calculate the slope of these tangent lines. These slopes represent the instantaneous velocity of the glider at t = 0.4 and 1.0 seconds. Use -~ m (VI,O_ o e |Ei|=m to calculate the glider's average acceleration on the graph, then transfer the result to Data Table Three as axr. Q5. Use a Cartesian coordinate system to graph the instantaneous velocity of the glider as a function of time. Place the vertical axis about one-third of the way from the left edge of the paper. Draw a best-fit line to your data and extend the line across the entire graph, all the way to the horizontal (time) axis . Q6. Note the point where your line crosses the velocity axis. This intercept is the initial velocity of the glider at the data point chosen by your instructor as t = 0. Record the value of v, on your graph. On the graph, calculate the slope of your best fit line and transfer the results to Data Table Three as avr. Q7. Use the data from Q6 to predict the position of the glider 1.0 seconds after release, using xf:0+\\?'0t+%a'{,rt2 . Compare your result to the position of the glider at t = 1.0 seconds from Data Table Two. Calculate the percent (difference or error, as appropriate) between the predicted and measured glider positions. We would prefer to use the raw position and time data, but it is not a straight line on a Cartesian scale. Any regular curve on a Cartesian scale can be represented as an exponential function. In our case we think that x = C (t)" will work, where 'C' is a constant and 'n' is the exponential dependence. We can use the property of logs to take our raw data and 'linearize' it. Take the log of both sides: log x=log(Ct")=logC +log(t")=log C+nlogt . This is of the form y = b + m x with b=1log C, m=nand log t = x. We turn the graph into a straight line by plotting it on a logarithmic scale. To make this technique simpler to use requires that we make sure that the glider has v = 0 when t = 0, instead of the arbitrary position chosen by your instructor. To correct your data for the delayed start time selected by the instructor, examine your velocity graph. The velocity line crosses the time axis at a 'negative time' or a 'time before zero'. This represents the time difference between when the glider was actually released and the arbitrary time chosen by your instructor. rev 1222 Physics 195: Linear Kinematics Page 4 of 6 Q8. Record the absolute value of this time difference as T on your velocity graph. Next, add T to each of the time values in Data Table Two and record the results in column five, under tc. Q9. During the 'time before zero', the glider moved from the release point. To find out how far it displaced from rest, use X(_:%a},.r2 . Add X to each of the position values in Data Table Two and record the results in column six, under Xc. Q10. Use a logarithmic scale for both axes and plot the xc vs tc values. When you plotted x vs t in Q3, the best-fit line was a curve. Now the data is linear. Draw a best fit line to the data and calculate the slope of the log-log line on the graph. Show your work on the log-log graph and round the slope to a single decimal place. Q11. With the value of 'n' known, use your graph to determine the value and units of 'C', then write the complete equation of motion of the glider in the form x = Ct". Q12. Use the value of C to determine the acceleration of the glider. Record the result in Data Table Three as aioc. Q13. Calculate the percent (difference or error, as appropriate) between each of the experimental results and the theoretical model. What are your conclusions regarding the validity of the theoretical model versus the experimental conditions? Q14. Calculate the percent (difference or error, as appropriate) between each of the experimental results. What are your conclusions regarding the precision of the experimental techniques for obtaining the glider acceleration? Physics 195: Linear Kinematics Page 6 of 6 Data Table Three Source Value cm $ 2 Theoretical Calculation (aMODEL) Data Table (aRAW) Position Graph (axT) Velocity Graph (avr) Logarithmic Position Graph (aLOG) rev 1222Data Table One D = 100 cm H = 10 cm Data Table Two Dot x (cm) t (s) Ax (cm) cm Xc (cm) to (s ) Vavg S 0 cm 0 3.2 cm 32 cm/s 3.2 cm 0.1 5.1 cm 51 cm/s 8.3 cm 0.2 0.3 cm 3 cm/s 8 cm 0.3 3.2 cm 32 cm/s 11.2 cm 0.4 3.1 cm 31 cm/s 6 14.3 cm 0.5 0.9 cm 9 cm/s 13.4 cm 0.6 0.8 cm 8 cm/s 12.6 cm 0.7 0.8 cm 8 cm/s 11.8 cm 0.8 0.6 cm 6 cm/s 10 11.2 cm 0.9 0.9 cm 9 cm/s 11 10.3 cm 1.0 0.8 cm 8 cm/s 12 9.5 cm 1.1 0.8 cm 8 cm/s 13 8.7 cm 1.2 0.8 cm 8 cm/s 14 7.9 cm 1.3 0.8 cm 8 cm/s 15 7.1 cm 1.4