Problem 52 - College Physics

Question

A woman starts from her house and begins jogging down her street. In order to keep a record of her workout, she carries a stopwatch and records the time it takes her to reach specific landmarks, such as intersections, school grounds, and churches. since she knows the distance to each of these landmarks, she is able to tabulate the data shown here: $$ \begin{array}{ll} \hline \text { Time } & \text { Distance jogged } \\ \hline 10 \mathrm{~min} & 2.1 \mathrm{~km} \\ 17 \mathrm{~min} & 3.2 \mathrm{~km} \\ 32 \mathrm{~min} & 6.5 \mathrm{~km} \\ 44 \mathrm{~min} & 8.6 \mathrm{~km} \end{array} $$ Make a plot of the distance jogged versus time. Draw a best-fit line through the data, and use this line to estimate (a) the speed at which she was jogging in \(\mathrm{m} / \mathrm{s}\) and \((\mathrm{b})\) the distance she will jog in 1 hour.

Step-by-Step Solution

Answer

(a) Average speed: 3.19 m/s; (b) Distance in 1 hour: 11.5 km.

Step 1: Prepare the data for plotting

We have the following data: \( (10, 2.1), (17, 3.2), (32, 6.5), (44, 8.6) \), where the first value is time in minutes and the second value is the distance in kilometers.

Step 2: Create the plot

Use a graphing tool or graph paper to plot the points on a coordinate plane. The x-axis will represent the time in minutes and the y-axis will represent the distance in kilometers. Plot the points: \((10, 2.1), (17, 3.2), (32, 6.5), (44, 8.6)\).

Step 3: Draw the best-fit line

By observing the plotted points, sketch a straight line that best follows the trend of the data points. This line may not pass through all points but should have roughly the same number of points above and below the line.

Step 4: Calculate the slope of the best-fit line

Select two points from the line to calculate the slope, which represents speed. For instance, use points \((10, 2.1)\) and \((44, 8.6)\). The slope \(m\) is given by:\[m = \frac{(8.6 - 2.1) \text{ km}}{(44 - 10) \text{ min}} = \frac{6.5}{34} \text{ km/min}\]Convert this speed to meters per second:\[m = \frac{6.5 \times 1000 \text{ m}}{34 \times 60 \text{ s}} \approx 3.19 \text{ m/s}\]

Step 5: Estimate distance in 60 minutes using the slope

Use the line’s equation to estimate the distance jogged in 60 minutes. With speed approximately \(3.19 \text{ m/s}\), convert this back to km for 60 minutes:\[\text{Distance} = 3.19 \times 60 \times 60 \div 1000 \approx 11.5 \text{ km}\]This estimation uses the assumption that the speed remains constant.

Key Concepts

Distance-Time Graph

When analyzing uniform motion, a distance-time graph is an essential tool. It helps visualize how distance varies with time. In this scenario, we plot time on the x-axis and distance on the y-axis. Using the provided data points

• (10, 2.1),
• (17, 3.2),
• (32, 6.5),
• (44, 8.6)

we create a graph to show this relationship. Each point represents a specific time and the corresponding distance jogged by the woman.

As you plot these points, you'll observe the increasing trend in distance over time. Such a graph allows us to identify patterns in the motion, making it easier to estimate future distances or find the speed of jogging.

Speed Calculation

Speed is calculated as the rate of change of distance with respect to time. In a distance-time graph, it directly corresponds to the slope of the line connecting data points. The slope can be calculated using the formula:

\[m = \frac{\Delta \, ext{Distance}}{\Delta \, ext{Time}}\] For instance, considering points (10, 2.1) and (44, 8.6), we find the change in distance (6.5 km) over the change in time (34 minutes):

\[m = \frac{6.5 \, ext{km}}{34 \, ext{minutes}} \approx 0.191 \, ext{km/minute}\]To convert this to meters per second, we multiply by 1000 (to convert km to meters) and divide by 60 (to convert minutes to seconds), resulting in a speed of approximately 3.19 m/s.

This technique highlights how simple calculations can provide meaningful insight into motion data.

Best-Fit Line

A best-fit line on a distance-time graph represents the overall trend of the data. It's a straight line that might not pass through all data points but balances the number above and below it. This helps identify the constant rate of change—or uniform speed.
To draw this line, look for the trend in your plotted points. Sometimes, tools are used to calculate this line mathematically, but a visual estimation works well in many cases. Selecting any two points on this line, we use them to calculate the slope, thus estimating the average speed.
The primary purpose of a best-fit line is to predict unknown values—for instance, estimating future distance jogged over a continuous stretch of time, based on present data. This approach assumes the motion remains uniform.

Unit Conversion

Unit conversion is crucial in ensuring uniformity in calculations, especially in physics. When working with speed, distances are often measured in kilometers or meters, whereas time can be in minutes or seconds. Converting between these units ensures accurate calculations.

• To convert kilometers to meters, multiply by 1000.
• To convert minutes to seconds, multiply by 60.

For example, converting a speed of 0.191 km/min to meters per second involves multiplying by 1000 and dividing by 60, yielding approximately 3.19 m/s.
Being comfortable with unit conversions allows you to tackle a wide range of problems effectively, providing results in usable formats that align with standardized scientific practices.