Problem 118 - Fundamentals of Physics

Question

The wings on a stoncfly do not flap, and thus the insect cannot fly. However, when the insect is on a water surface, it can sail across the surface by lifting its wings into a hreere. Suppose that you time stoneflies as they move at constant speed along a straight path of a certain length. On average, the trips each take \(7.1 \mathrm{~s}\) with the wings set as sails and \(25.0 \mathrm{~s}\) with the wings tucked in. (a) What is the ratio of the sailing speed \(v_{s}\) to the nonsailing speed \(v_{\sin } ?\) (b) In terms of \(v_{s .}\) what is the difference in the times the inscets take to travel the first \(2.0 \mathrm{~m}\) along the path with and without sailing?

Step-by-Step Solution

Answer

(a) The ratio is \(\frac{25.0}{7.1}\). (b) The time difference is \((2.0/d) \times 17.9\).

Step 1: Define the Variables

Let the distance traveled by the stoneflies be denoted as \(d\). The sailing speed is \(v_{s}\) and the non-sailing speed is \(v_{ns}\). We know the time taken to travel with sailing is \(t_s = 7.1\, \text{s}\) and without sailing is \(t_{ns} = 25.0\, \text{s}\).

Step 2: Find the Sailing and Non-Sailing Speeds

We use the formula for speed: \(v = \frac{d}{t}\). Thus, the sailing speed \(v_{s} = \frac{d}{7.1}\) and the non-sailing speed \(v_{ns} = \frac{d}{25.0}\).

Step 3: Calculate the Ratio of Speeds

The ratio of the sailing speed to the non-sailing speed is given by \(\frac{v_{s}}{v_{ns}} = \frac{\frac{d}{7.1}}{\frac{d}{25.0}}\). Simplifying this gives \(\frac{v_{s}}{v_{ns}} = \frac{25.0}{7.1}\).

Step 4: Calculate the Time Difference for 2.0 Meters

First, find the time with sailing for 2.0 meters: \(t_{s,2.0} = \frac{2.0}{v_{s}}\). Substituting \(v_s = \frac{d}{7.1}\), we have \(t_{s,2.0} = \frac{2.0 \times 7.1}{d}\). Next, find the time without sailing for 2.0 meters: \(t_{ns,2.0} = \frac{2.0}{v_{ns}} = \frac{2.0 \times 25.0}{d}\). The difference in time \(\Delta t = t_{ns,2.0} - t_{s,2.0} = \frac{2.0 \times 25.0}{d} - \frac{2.0 \times 7.1}{d}\). Simplifying gives \(\Delta t = \frac{2.0}{d} (25.0 - 7.1)\).

Key Concepts

Speed

Speed is a fundamental concept in kinematics. It is defined as the distance traveled by an object per unit time. In mathematical terms, speed can be calculated using the formula \( v = \frac{d}{t} \). Here, \( v \) represents speed, \( d \) is the distance covered, and \( t \) is the time taken to cover that distance.
In our exercise, stoneflies travel over a certain path while taking two distinct approaches - sailing and non-sailing. Each method results in a different speed due to varying times, showing that the shorter the time for a given distance, the higher the speed.
The sailing speed \( v_s \) of the stonefly is noticeably higher because it navigates the path faster within 7.1 seconds as opposed to 25 seconds when not using their wings as sails. Therefore, understanding speed in this context helps us realize how quickly the stoneflies can cross a specific distance based on their mode of travel.

Ratio of speeds

The ratio of speeds provides a comparison between two different speeds. It helps to understand how much faster or slower one speed is in relation to the other. This is especially useful when looking at objects or organisms like stoneflies that can move in multiple ways.
In the given exercise, the ratio of sailing speed \( v_{s} \) to non-sailing speed \( v_{ns} \) is determined by comparing their respective speeds using the formula from their respective times. The calculation \( \frac{v_{s}}{v_{ns}} = \frac{25.0}{7.1} \) reflects how sailing is significantly faster than non-sailing. These kinds of ratios offer a clear view of the performance differences in speed and allow for easier comparison when making practical decisions or evaluations about movement efficiency.

Time difference

Time difference is a critical concept when assessing how different conditions affect the time it takes to cover a certain distance. If you are comparing two methods of travel, understanding the time difference can tell you which method is quicker and by how much.
In the stonefly scenario, we calculate the time taken to cover a distance of 2.0 meters both with and without sailing. This involves using the already determined speeds to see the times for each scenario. The time with sailing for 2.0 meters involves \( t_{s,2.0} = \frac{2.0 \times 7.1}{d} \), and without sailing, it is \( t_{ns,2.0} = \frac{2.0 \times 25.0}{d} \).
The time difference \( \Delta t \) between these two travel methods is found by subtracting the time taken by sailing from the time taken without sailing as demonstrated by the formula \( \Delta t = \frac{2.0}{d} (25.0 - 7.1) \). This gives a clear indication of how much longer one method takes than the other, which is particularly useful in optimizing travel methods for speed.