COMPARING KIETIC ENERGIES
Two iceboats like the one in Example 5.6 (Section 5.2) hold a race on a frictionless horizontal lake (Fig. 6.14). The two iceboats have masses m and 2m. The iceboats have identical sails, so the wind exerts the same constant force \overrightarrow{\boldsymbol{F}} on each iceboat. They start from rest and cross the finish line a distance s away. Which iceboat crosses the finish line with greater kinetic energy?
If you use the definition of kinetic energy, K=\frac{1}{2} m \bar{v}^{2}, Eq. (6.5), the answer to this problem isn’t obvious. The iceboat of mass 2m has greater mass, so you might guess that it has greater kinetic energy at the finish line. But the lighter iceboat, of mass m, has greater acceleration and crosses the finish line with a greater speed, so you might guess that this iceboat has the greater kinetic energy. How can we decide?
The key is to remember that the kinetic energy of a particle is equal to the total work done to accelerate it from rest. Both iceboats travel the same distance s from rest, and only the horizontal force F in the direction of motion does work on either iceboat.
Hence the total work done between the starting line and the finish line is the same for each iceboat, W_{tot} = Fs. At the finish line, each iceboat has a kinetic energy equal to the work W_{tot} done on it, because each iceboat started from rest. So both iceboats have the same kinetic energy at the finish line!
You might think this is a “trick” question, but it isn’t. If you really understand the meanings of quantities such as kinetic energy, you can solve problems more easily and with better insight.
Notice that we didn’t need to know anything about how much time each iceboat took to reach the finish line. This is because the work–energy theorem makes no direct reference to time, only to displacement. In fact the iceboat of mass m has greater acceleration and so takes less time to reach the finish line than does the iceboat of mass 2m.