Uniform Accelerated Motion (1)

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Title of the Experiment: Constant Acceleration Linear Motion Student’s name: Anjana Shyam Section SLN: A TA’s Name: Ayush Kumar Singh Week of the experiment: 2

OBJECTIVE (3 points) : The purpose of this lab is to determine the relationship between position vs. time and velocity vs time graphs with motion of uniform acceleration. The experimental value of the acceleration of gravity will be determined through simulations and kinematic equations. EXPERIMENTAL DATA (3 points): Obtain experimental data that will be used for further calculations from the graphs. PART 1: Uniform Accelerated Motion on a Dynamic Track Table 1. Run 1a Time interval (units) Coordinates (units) Distance (units) 1s-2s 0.0771 m, 0.2824 m 1.02085 m 3s-4s 0.8297 m, 1.7155 m 1.3359 m After plotting a curve of best fit for x(t) in Logger Pro, fill in these tables with the corresponding curve fit coefficients. Pay close attention to the direction of the velocity and acceleration. Table 2. Position vs. Time Curve Fit Coefficients Run # A B C Acceleration (units) 1a 0.1702 -0.3047 0.2112 0.3406 m/s^2 1b 0.1334 -0.2248 0.1671 0.2668 m/s^2 2 -0.1686 0.2433 1.842 -0.3372 m/s^2 3 -0.0642 0.8129 -0.5485 -0.1284 m/s^2 4 0.2147 -1.828 4.034 0.4294 m/s^2 Table 3. Position vs. Time Curve Fit Parameter Definitions Coefficients Name of Physics quantity (i.e. position, distance, velocity, etc.) A ½ acceleration B Initial velocity C Initial position

After plotting a line of best fit for v(t) in Logger Pro, fill in the following tables with the corresponding linear fit parameters. Table 4. Velocity vs. Time Linear Fit Parameters Run # Slope Y-intercept Acceleration (units) 1a 0.3402 -0.3044 0.3402 m/s^2 1b 0.2718 -0.225 0.2718 m/s^2 2 -0.3398 0.2501 -0.3398 m/s^2 3 -0.2642 1.174 -0.2642 m/s^2 4 0.4293 -1.829 0.4293 m/s^2 Table 5. Velocity vs. Time Linear Fit Parameter Definitions Coefficients Name of Physics quantity (i.e. position, distance, velocity, etc.) m (slope) acceleration b (y-intercept) Initial velocity Run1: Cart speeds up while moving away from sensor

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PART 2: Free Fall After recording data from the video and fitting a curve of best fit to the x(t) graph and line of best fit to the v(t) graph, fill in the tables below. Table 6. Position vs. Time graph Curve Fit Parameters

A B C gravitational acceleration (units) -5.369 21.21 -13.5 -10.738 m/s^2 Table 7. Velocity vs. Time graph Linear Fit Parameters Slope Y-intercept gravitational acceleration (units) -10.5 20.58 -10.5 m/s^2 DATA ANALYSIS (10 points): the section includes sample calculations and error analysis. Be sure to include equations! PART 1: Uniformly accelerated motion along the dynamic track. Show how the acceleration of the cart was calculated using the coefficients of quadratic fit. (Refer to Table 2). Kinematics Equation: x f = x 0 + v 0 t +1/2at^2 Quadratic fit equations: Y = At^2 + Bt+ C A= 1/2a 0.1702 *2 =0.3404 0.1334*2= 0.2668 -0.1686*2 = -0.3372 -0.0642*2= -0.1284 0.2147*2= 0.4294 Write equations of the motion (position as a function of time) for the cart for each of the run, using the parameters in Table 2. Run 1a: X(t) = 0.2112-0.3047t+0.1702t^2 Run 1b: X(t) = 0.1334-0.2248t+0.1671t^2 Run 2 X(t)= -0.1686 +0.2433t+1.842t^2 Run 3: X(t) = -0.0642 + 0.8129t-0.5485t^2 Run 4: X(t) = 0.2147 -1.828t+4.034t^2 Write equations of the velocity for the cart for each of the run, using the parameters in Table 4 V(t) = V0 + a(t) Run 1a: v(t)= 0.3402t -0.3044 Run 1b: v(t) = 0.2718t -0.225 Run 2: v(t) = -0.3398t+0.2501 Run 3: v(t) = -0.2642t+1.174 Run 4: v(t) = 0.4293t-1.829

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PART 2: Free Fall Run A: Calculate the percent discrepancy between accepted value of gravitational acceleration and your experimental result: |g exp – g theor |/(g theor ) * 100% |10.738 – 9.8|/9.8 * 100 = 9.57% |10.5-9.8|/9.8*100= 7.14% Run B: From the average (mean) time of fall, calculate the experimental gravitational acceleration. Show equations and calculations. A=-2y/t^2 t=1.33 -6.48*2=-12.96 -12.96/(1.33^2)= -7.3265 m/s^2 Knowing that the uncertainty in height measurement is ∆H = 0.02 m and the average human response time is ∆t = 0.2 s, estimate the uncertainty (∆g ) in your experimental g. Show equations and calculations. (∆g/g)^2=(∆H/H)^2+(2∆t/t ̅ )^2 (0.02/6.48)^2+(2*0.2/1.33)^2 = 0.09046121 Sqrt(0.09046121)= 0.30076771 0.30076771*-7.3265~ -2.20 Calculate the final velocity (eq. 2) of the ball at the bottom of the fall and its uncertainty. (Hint: ((∆V_f)/V_f )^2 =(∆g/g)^2+(∆t/t ̅ )^2 ) 7.3265*1.33= 9.744245 m/s 0.09046121+ (0.2/1.33)^2= 0.11307414124 0.11307414124^2 = delta(vf)/9.744245 Uncertainty vf ~ 0.12 RESULTS (3 POINTS)

Note: Read the rules of the number significant figures to report in the final results. (pg. 6 Lab Manual). PART 1: Acceleration (units) Run # Position vs. Time graph Velocity vs. Time graph Write equation of motion for each run 1a 0.34 m/s^2 0.340 m/s^2 0.21-0.3t+0.17t^2 1b 0.27 m/s^2 0.27m/s^2 0.13-0.23t+0.17t^2 2 -0.34 m/s^2 -0.34 m/s^2 -0.17 +0.24t+1.84t^2 3 -0.13 m/s^2 -0.26 m/s^2 -0.06 + 0.81t-0.55t^2 4 0.43 m/s^2 0.43 m/s^2 0.21-1.83t+4.03t^2 PART 2: Free Fall Part: (gravitational acceleration ± uncertainty ) (units) Final velocity ± uncertainty ) (units) 2A 10.738±9.57 N/A 2B 10.5±7.14 9.744245±0.12 DISCUSSION & CONCLUSION (10 points): The purpose of this experiment was to experimentally determine the acceleration of gravity based on dropping a ball from a certain height, and to verify kinematics equations through the determining the graphical relationship between position vs. time graphs and velocity vs. time graphs. The coefficients of a quadratic equations that was used to approximate the position time graphs corresponded to the kinematic equation that is used to determine displacement from initial velocity, time duration, and acceleration. A linear fit was performed on velocity vs time graph, to determine the acceleration which was the slope. The linear fit corresponds to the kinematic equations used to determine final velocity by a given initial velocity, time duration, and acceleration. The acceleration determined from the velocity graph and the acceleration determined from the coefficients in the position-time graph equation were nearly identical, supporting the theoretical relationship between displacement, velocity, and acceleration. Since acceleration and velocity are vector quantities, when the signs don’t match the cart is slowing down, and when both signs match the cart is speeding up. Free fall motion has no initial velocity and a constant acceleration: the acceleration due to gravity. Human response time contributes a larger component to the overall error component of the experimental value of the acceleration due to gravity. The theoretical value of g fell within a reasonable range of the experimental value, however sources of error such as possible air resistance and discrepancies in reporting equal position-time interval could have caused the degree of experimental error in this lab. Air resistance opposes the force of gravity, which is impossible to replicate without, and the position- time and velocity-time intervals that were recorded weren’t exactly equal.

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