TOUCHSTONE EXAMPLE 2-5: Motion Data

Suppose that you gave a box sitting on a carpeted floor a push and then recorded its position three times per second as it slid to a stop. The table gives the results of such a measurement. Let's analyze the position vs. time data for the box sliding on the carpet and use curve fitting and calculus to obtain the velocity measurements. We will use Excel spreadsheet software to perform our analysis, but other computer- or calculator-based fitting or modeling software can be used.

(a) Draw a graph of the x vs. t data and discuss whether the relationship appears to be linear or not.

SOLUTION: The Key Idea here is that the relationship between two variables is linear if the graph of the data points lie more or less along a straight line.There are many ways to graph the data for examination: by hand, with a graphing calculator, with a spreadsheet graphing routine, or with other graphing software such as Data Studio (available from PASCO scientific) or Graphical Analysis (available from Vernier Software and Technology). The graph in Fig. 2-18 that shows a curve and so the relationship between position, x, and time is not linear.

Figure 2-18 Solution to Touchstone Example 2-5(a). A graph of position versus time for a box sliding across a carpet.

. (b) Draw a motion diagram of the box as it comes to rest on the carpet.

SOLUTION: The Key Idea here is to use the data to sketch the position along a line at equal time intervals. In Fig. 2-19, the black circles represent the location of the rear of the box at intervals of 1/30 of a second.

Figure 2-19 Solution to Touchstone Example 2-5(b). A motion diagram for a box sliding across a carpet.

. (c) Is the acceleration constant? If so, what is its component along the x axis?

SOLUTION: The Key Idea here is to explore whether or not the relationship between position and time of the box as it slides to a stop can be described with a quadratic (parabolic) function of time as described in Eq. 2-17. This can be done by entering the data that are given into a spreadsheet or graphing calculator and either doing a quadratic model or a fit to the data.The outcome of a quadratic model is shown in Fig. 2-20. The x-model column contains the results of calculating x using the equation for each of the times in the first column using the initial position, velocity and acceleration data shown in the boxes. The line shows the model data. If the kinematic equation fits the data, then we can conclude that the acceleration component is a constant given by a_x = –6.6 m/s². Thus the acceleration is in the negative y direction.

Figure 2-20 Solution to Touchstone Example (c). Data and a graph of position as function of time for a box sliding over carpet. Actual data is compared to a model of what is expected from Eq. 2-17 (assumed constant acceleration).The value of acceleration which produced the best match between the model and actual data is –6.6 m/s².