Question

A displacement boat (generally a moving object held up by buoyancy of the water) generates a wave that travels alongside at the speed of the boat with the first wave crest at the bow of the boat. Because the wavelength increases with the boat's speed, above a certain speed the second crest appears astern of the boat, so that the boat must travel uphill on the wave it creates. If it lacks the power to get over the top of this wave, its speed is limited to roughly the speed of a wave with the length of the boat. Hence we have the following rule of thumb:

The maximum speed of a modestly powered boat is approximately equal to (actually, a little faster than) the wave whose wavelength equals the length of the boat measured at the waterline.

Empirically, the propagation velocity of water waves is

v=(g/k)^1/2

where k is the wavenumber and g is the gravitational constant. In parts that ask for numerical answers, use g=9.81m/s2.

1) Find the velocity of water waves v in terms of the wavelength lambda.

Express the velocity in terms of lambda and constants like g and pi.

An eight-person shell (or "eight") is a light rowing craft approximately 17.6-m long with eight rowers and a coxswain who steers and calls the stroke. Assume that the speed of the shell is limited according to the rule given in the introduction.

2) Estimate the time T8 it takes an "eight" to travel 2000 m.

Express the time numerically, in seconds, to three significant figures.

3) Estimate the time T100 it takes a good human swimmer, 1.80-m tall, to swim 100 m. Assume that the speed of the swimmer is limited by his or her height.

Express the time numerically, in seconds, to three significant figures.

Answer 1