Halliday, Resnick & Walker — Potential Energy and Conservation of Energy: Problem 136

Fly-fishing and speed amplification. If you throw a loose fishing fly, it will travel horizontally only about 1 m. However, if you throw that fly attached to a fishing line by casting the line with a rod, the fly will easily travel horizontally to the full length of the line, say, 20 m. The cast is depicted in Fig. 8.58: Initially (Fig. 8.58a) the line of length \( L \) is extended horizontally leftward and moving rightward with speed \( v_0 \). As the fly at the end of the line moves forward, the line doubles over, with the upper section still moving and the lower section stationary (Fig. 8.58b). The upper section decreases in length as the lower section increases in length (Fig. 8.58c), until the line is extended rightward and there is only a lower section (Fig. 8.58d). If air drag is neglected, the initial kinetic energy of the line in Fig. 8.58a becomes progressively concentrated in the fly and the decreasing portion of the line that is still moving, resulting in an amplification (increase) in the speed of the fly and that portion. (a) Using the x axis indicated, show that when the fly position is \( x \), the length of the still-moving (upper) section of line is \( (L - x)/2 \). (b) Assuming that the line is uniform with a linear density \( \rho \) (mass per unit length), what is the mass of the still-moving section? Next, let \( m_f \) represent the mass of the fly and assume that the kinetic energy of the moving section does not change from its initial value (when the moving section had length \( L \) and speed \( v_0 \)) even though the length of the moving section is decreasing during the cast. (c) Find an expression for the speed of the still-moving section and the fly. Assume that initial speed \( v_0 = 6.0 \text{ m/s} \), line length \( L = 20 \text{ m} \), fly mass \( m_f = 0.80 \text{ g} \), and linear density \( \rho = 1.3 \text{ g/m} \). (d) Plot the fly's speed \( v \) versus its position \( x \). (e) What is the fly's speed just as the line approaches its final horizontal orientation and the fly is about to flip over and stop?