Question

The wings on a stonefly 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 breeze. 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$\mathbf{7}\mathbf{.}\mathbf{1}\mathbf{}\mathbf{s}$with the wings set as sails and $\mathbf{25}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{s}$with the wings tucked in. (a) What is the ratio of the sailing speed ${\mathbf{v}}_{\mathbf{s}}$to the non-sailing speed localid="1654754209433" ${\mathbf{v}}_{\mathbf{ns}}$? (b) In terms oflocalid="1654754226410" ${\mathbf{v}}_{\mathbf{s}}$, what is the difference in the times the insects take to travel the first$\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}$along the path with and without sailing?

Short answer

(a)The ratio of the sailing speed ${\mathrm{v}}_s$to the non-sailing speed ${\mathrm{v}}_{\mathrm{ns}}$is $3.52$.

(b)The difference in the times to travel for the first 2 meters along the path with sailing and without sailing is$5.04/{\mathrm{v}}_{\mathrm{s}}$

Step-by-step solution

Given data

Time duration for the sailing,$t_{\mathrm{s}}=7.1\mathrm{s}$

Time duration for the non-sailing,$t_{\mathrm{ns}}=25.0\mathrm{s}$

Understanding the average speed

Average speed can be found by dividing the total distance by the total time. In this case, the total distance traveled by the stoneflies would be the same in both cases.

The expression for the average speed is given as follows:

${\mathrm{v}}_{\mathrm{avg}}=\frac{\mathrm{x}}{\mathrm{t}}$ … (i)

Here,$x$is the total distance and $t$is the total time.

(a) Determination of the ratio of sailing to non-sailing speeds

Since, total distance is same in both the cases therefore from equation (i),

$$\begin{gathered} \frac{{\mathrm{v}}_{\mathrm{s}}}{v_{ns}}=\frac{x}{t_s}\times \frac{t_{ns}}{x} \\ =\frac{t_{ns}}{t_s} \end{gathered}$$

Substitute the values in the above expression.

$$\begin{gathered} \frac{{\mathrm{v}}_{\mathrm{s}}}{v_{ns}}=\frac{25.0\mathrm{s}}{7.1\mathrm{s}} \\ =3.52 \end{gathered}$$

Thus, the ratio of the sailing speed ${\mathrm{v}}_s$to the non-sailing speed${\mathrm{v}}_{\mathrm{ns}}$of stoneflies is$3.52$.

(b) Determination of the difference in time in the two cases

The relation between sailing speed and non-sailing speed is,

${\mathrm{v}}_{\mathrm{ns}}=\frac{{\mathrm{v}}_{\mathrm{s}}}{3.52}$

Time for traveling 2 m with sailing,$t_s=\frac{2}{v_s}$

Time for traveling 2 m without sailing,$t_s=\frac{2}{v_{ns}}$

Difference between the times would be,

$$\begin{gathered} ∆t=t_{ns}-t_s \\ =\frac{2}{v_{ns}}-\frac{2}{v_s} \\ =\frac{2\times 3.52}{v_n}-\frac{2}{v_n} \\ =\frac{5.04}{v_n} \end{gathered}$$

Thus, the difference between the time for travel$2\mathrm{m}$with sailing and without sailing is$5.04/{\mathrm{v}}_{\mathrm{s}}$.