Lab 2：Acceleration of Gravity

Purpose: 1. To determine the accelaration.
                2. To gain experience using the computer as a data collector.

Equipment: Windows based computer, Lab Pro interface, Logger Pro Software, motion detector, rubber         ball, wire basket.

Introduction: In this laboratory you will use the computer to collect some position (x) vs time (t) data for a rubber ball tossed into air. Since the velocity of an object is equal to the slope of the x vs t curve, the computer can also construct the graph of v vs t by calculating the slope of x vs tat each point in time. We will  use both the x vs t graph and the v vs t graph to find the free fall acceleration of the ball.

Part 1:

  (1) Position vs. time graph:

       position (x) [m] vs. time (t) [s] graph

  (2) Function:

      We selected an appropriate data range and tried to find a function that fit the curve. 

      We chose the Quadratic (At^2+Bt+C), our equation is "x = -4.858t^2 + 8.535t - 2.163".

      (A = -4.858,  B = 8.535,  C = -2.163)

  (3) Slope:

      Because "velocity = (△x) / (△t)" , which is the derivative of the position (x) vs. time (t) curve.
      So the slope of the  position vs. time curve is the velocity.

  (4) Velocity:

      According to the graph, we found that between 0.5s and 0.85s the slope is positive, if we assume the upward direction is positive direction, velocity between 0.5s and 0.85s is positive; the slope between 0.85s and 1.33s is negative, so the velocity is also negative.

  (5) Acceleration:

     For a linear motion with constant acceleration: during the time interval "△t = tf - ti " ,

     sf = si+vis△t+(1/2)as(△t)^2 ——  note that "sf = final position", "si = initial posotion", 

                                                                      "vis=initial velocity", "as = accelaration".

   
     So "A" in our quadratic equation is equal to (1/2) as .      

     So  acceleration=2A=2 x (-4.858)= -9.716m/s^2

  

  (6) Gravity:   

     We assume that gravity is equal to acceleration.
     So our "gexp =2A=2 x (-4.858) =-9.716m/s^2 ".

     (Accepted value of gravity: gacc= -9.8m/s^2)

  (7) Error: 

     Percent error = [(measured- actual) / (actual)] x 100%  

                          =  [(9.8m/^2-9.716m/^2) / (9.8m/s^2)] x 100% = 0.92% .

     That means our data is really close to the accepted value.

Part 2:

     

 (1) Velocity vs. time graph:

    velocity (v) [m/s] vs. time (t) [s] graph

  
  (2) Function: 

          We selected an appropriate data range and tried to find a function that fit the curve. 

          We chose the linear function (v=mt+b), our equation is "v = -9.799t + 8.613".

          (m = -9.799, b = 8.613)

   
  (3) Slope:

         Because "acceleration = (△v) / (△t)" , which is the derivative of the velocity (v) vs. time (t) curve.
         So the slope of the velocity vs. time curve is the acceleration.

   
  (4) Acceleration:

         Because  the slope (m) of the velocity vs. time curve is the acceleration. 

         So the "acceleration = m = -9.799 m/s^2".

  (5) Gravity:

        We assume that gravity is equal to acceleration, so our "gexp = m = -9.799 m/s^2". 

        (Accepted value of gravity: gacc= -9.8m/s^2) 

 
  (6) Error: 

       Percent error = [(measured- actual) / (actual)] x 100%  

                          =  [(9.8m/^2-9.799m/^2) / (9.8m/s^2)] x 100% = 0.01% .

       That means our data is really close to the accepted value.

Then we repeated our experiment for several times to get the average data:

Conclusion:

       In this lab, we determine the gravity is close to 9.8m/s^2.
       In order to get the most precise data, we did this experiment for many times and got the average data, which decrease the error accidental error. Our data is 3.18% varying from the actual, that mean this experiment can prove that the gravity for a freely falling object is close to 9.8m/s^2. Our errors are because of :
      1. Air resistance
      2. The inevitable experimental error, because the equipment can't be exactly precise.
      3. The curve fit is an estimate, so our gravity is also an estimate value.

      We also learned how to use Lab Pro interface, Logger Pro Software, motion detector and gained the experience using the computer as a data collector. We also worked as a team to gain and analysis the data. This experience may benefit us in the future.