Question

A flea hops in a straight path along a meter stick, starting at \(0.7 \mathrm{~cm}\) and making successive jumps, which are measured to be \(3.2 \mathrm{~cm}, 6.5 \mathrm{~cm}, 8.3 \mathrm{~cm}, 10.0 \mathrm{~cm}, 11.5 \mathrm{~cm}\) and \(15.5 \mathrm{~cm} .\) Express the answers to the following questions in scientific notation, with units of meters and an appropriate number of significant figures. What is the total distance covered by the flea in these six hops? What is the average distance covered by the flea in a single hop?

Short answer

Answer: The total distance covered by the flea in these six jumps is \(5.5 \times 10^{-1}~m\), and the average distance covered by the flea in a single hop is \(9.17 \times 10^{-2}~m\).

Step-by-step solution

Convert distances to meters

First, we need to convert all distances (in centimeters) to meters by multiplying them by a conversion factor of 0.01. Initial position: \(0.7~cm \cdot 0.01 = 0.007~m\) Jumps: \(3.2~cm \cdot 0.01 = 0.032~m\) \(6.5~cm \cdot 0.01 = 0.065~m\) \(8.3~cm \cdot 0.01 = 0.083~m\) \(10.0~cm \cdot 0.01 = 0.1~m\) \(11.5~cm \cdot 0.01 = 0.115~m\) \(15.5~cm \cdot 0.01 = 0.155~m\)

Find the total distance covered by the flea

Add up the distances of all six jumps: Total distance: \(0.032~m + 0.065~m + 0.083~m + 0.1~m + 0.115~m + 0.155~m = 0.55~m\) Express the total distance in scientific notation: \(5.5 \times 10^{-1}~m\)

Calculate the average distance covered by the flea in a single hop

Divide the total distance by the number of hops (6 jumps) to obtain the average distance per hop: Average distance: \(\frac{0.55~m}{6} = 0.0916666667~m\)

Express the average distance in scientific notation with appropriate significant figures

Round the average distance to three significant figures and express it in scientific notation: Average distance: \(9.17 \times 10^{-2}~m\) The total distance covered by the flea in these six hops is \(5.5 \times 10^{-1}~m\), and the average distance covered by the flea in a single hop is \(9.17 \times 10^{-2}~m\).

Key concepts

Unit Conversion

When tackling physics problems involving measurements, unit conversion is a fundamental skill to master. It is the process of converting a quantity expressed in one unit of measurement into another unit of measurement, while keeping the actual size or value of the quantity unchanged.

Let's dive into the provided physics problem: A flea's hopping distance is given in centimeters, which is a standard unit of length in the metric system. However, in the world of scientific research and reporting, distances are often represented in meters, which is the SI base unit for length. To convert centimeters to meters in our flea example, we multiply the distances by the conversion factor of 0.01, as there are 100 centimeters in a meter.

Understanding this conversion factor is critical in ensuring accurate calculations and consistency in the units used throughout scientific notation physics problems. Keep in mind that incorrect unit conversion can lead to enormous errors in real-world scenarios including scientific research and engineering projects.

Significant Figures

Significant figures, or 'sig figs' as they are sometimes called, are a way of expressing the precision of measurements. In physics problems, they indicate how many digits can be reliably used from a measurement when making calculations.

In the flea hopping problem, converting measurements to scientific notation also involves rounding them to an appropriate number of significant figures. This tells us not only the order of magnitude but also the precision of the flea's jumps. At this point, it's important to notice that a measurement like '10.0 cm' has three significant figures, which suggests it's more precise than a measurement of '10 cm', which would imply just one significant figure.

Our final answer needs to respect these levels of precision. For instance, expressing the average jump distance as \(9.17 \times 10^{-2}~m\) relates to the accepted significant figures obtained from the original measurements. Accurately determining significant figures is crucial for reporting measurements in scientific work, and it's a skill highly valued in professional settings as well.

Average Distance Calculation

Calculating an average is a staple mathematical operation, and in physics problems, it often represents an overall trend or a consistent measure of an entity. The average distance calculation provides a single value that represents the typical extent of a given set of distances.

For the flea example, we find the average distance covered by simply summing all the individual hops and then dividing by the total number of hops. Although the flea's jumping distances vary, calculating their average gives us a helpful indication of the flea's typical leap length.

This concept of averages extends beyond simple motion problems, and it's a critical part of data analysis across scientific disciplines. Whether measuring the average speed of a galaxy or examining the mean temperature of a chemical reaction, calculating averages allows us to make sense of repeated observations, even when those observations have natural variations.