Linearizing Data - Fence Lab

Using the BALLDROP lab, we know that linearization does not work well when the data does not begin when time t=0 seconds. The objective of this page is to demonstrate what happens when the lab starts when t = 0 seconds and there is an initial velocity.

Setup

This lab involves dropping a vernier picket fence vertically through a photogate to measure the acceleration due to gravity. A program named FENCE was used, but the same data may be recorded using the vernier programs TIMER which comes in the group of files PHOTOGATE.

Data

The picket fence had a width of 5 cm or 0.05 m. The time was recorded using the CBL. The data is shown in Table 1.

A Mathematical Analysis

Analyzing the data in a similar fashion to the linearization of a dropped ball, the graph of the data in Image 1 appears to be in the relationship y = k*x^2. Using the quadratic regression provided on the TI-8? series calculator, the equation y = 4.90x^2 + 0.64x. Note that -4.31 x 10^(-5) is nearly zero. Recall that the physics formula for vertical displacement is d = (1/2)*a*t^2 + v_i*t + d_i where "a" is the acceleration, "v_i" is the initial velocity and "d_i" is the initial height of the object. The regression equation is d = 4.90t^2 + 0.64t + 0. (Note: The CBL arbitrarily measured the distance as a positive when the fence dropped toward the floor resulting in a positive value for the acceleration due to gravity.) Interpreting the data, we find 1/2 of the acceleration is 4.90 m/s^2. To find the acceleration, multiply this value by 2 and the acceleration of the ball is 9.798 m/s^2 (theoretical value 9.8 m/s^2, % error = 0.02%). The initial velocity is 0.64 m/s with an initial distance of 0 m. Since the picket fence was held above the photogate and dropped, there is a small initial velocity but no initial height as the timing mechanism was triggered when the fence broke the photogate.

Image 1	Image 2

The graph of the quadratic regression equation is shown with the data in Image 3. When the completing the square method is used, the equation can be converted from quadratic form [ y = ax^2 + bx + c] to parabolic form [y = a(x-h)^2 + k, vertex is (h,k)]. Converting to the variables d & t, parabolic form would be d = 4.90(t - -0.065)^2 - 0.021. The h value of the vertex represents the time at which the vertex occurs and the k value indicates the height at which the vertex occurs. Again, remember that the data was recorded so that distance toward the floor is positive. Analyzing the data in parabolic form indicates that the picket fence was released from a height 0.021 m above the photgate and 0.065 seconds prior to tripping the photogate. Hence we know that the vertex of the parabola is very close to (0 s, 0 m). The conclusion from the DROPBALL lab indicates that linearization does not work well if the time at the vertex of the parabola is not zero.

Linearizing the data

Since the graph appears to be distance vs. time^2, to linearize the graph one must square the time and graph d vs t^2. The data for this graph is shown in Table 2. The linearized graph is shown in Image 5 with the regression equation in Image 6.

Image 5	Image 6

The graph of the linear regression equation with the linearized data is shown in Image 7. Immediately, one notices that the y-intercept of the graph is not zero. Converting the equation y = ax + b to the values following linearization, the equation would be d = 7.63t^2 + 0.02 (graphed in Image 8). Once again, the 7.63 m/s^2 is expected to be 4.9 m/s^2 (1/2 of the acceleration due to gravity). The error is 56%. The y-intercept is at 0.02 m. This distance is 5.7% of the maximum distance of the picket fence timed by the CBL. Hence, if one adheres to the 5% rule, it cannot be considered zero.

Image 7	Image 8

The conjecture for the error in linearization involves the fact that the linearization method converts the equation to d = m*t^2 + b form. Recall the complete form of vertical motion of an object is d = (1/2)a*t^2 + v_i*t + d_i where "a" is the acceleration due to gravity, "v_i" is the initial velocity, and "d_i" is the initial height. Linearization assumes that the initial velocity must be zero as there is no "v_i * t" term in the equation. Since the timing mechanism used in this experiment recorded that time at zero, the discrepancy may result from the object being dropped from a small distance above the photogate resulting in a small initial velocity which results in a significant error in the calculation of the acceleration using the linearization method.

A Third Method - Change in Velocity / Change in Time

Acceleration is defined to be the change in velocity divided by the change in time. Using the data gathered from the CBL as shown in Table 1, it is possible to calculate the change in velocity.

Table 2 shows the calculation of velocity using the distance and time. Since velocity equals distance divided by time, the change in velocity is the change in distance divided by the change in time. The average velocity is computed by the formula (t1 + t2)/2 while the velocity is calculated by the formula (d2 - d1)/(t2-t1).

Image 9 shows the graph of the Velocity (m/s) vs Average Time (s). The slope of the line shown in Image 10 is 9.83 m/s^2 or an error of 0.3%. The y-intercept of the graph is 0.632 m/s. Since the y-axis in this graph is velocity measured in m/s, this is the velocity of the fence at time t=0. Note that the initial velocity in the quadratic regression was calculated to be 0.637 m/s. These two calcuations confirm the accuracy and the use of the quadratic regression method to determine the acceleration and initial velocity.

Image 9	Image 10

Conclusion

The data gathered by the CBL when the picket fence is dropped through a photogate shows errors of less than 1% when analyzed using two different methods other than linearization. One possible explanation for the error in the linearization method is the fact that the picket fence has a small but significant initial velocity when it passes through the photogate and begins the timing of the experiment.

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