Energy (also called work in physics) cannot be created or destroyed, it can only be transformed from one type to another.
There are two main types of mechanical energy; potential and kinetic.
Potential energy is energy stored internally in an object. A typical example is the energy stored in a compressed spring, or the energy something has by virtue of being lifted a certain height above the ground. In both these cases the potential energy is equal to the applied force multiplied by the distance over which it is applied, i.e.:
\begin{equation} E = fd \end{equation}In the case of the compressed spring, the force is the average "push" of the spring from its starting position to its compressed position, while the distance is the distance over which it is compressed. For gravitational potential energy, the force is the weight of the object, and the distance is the difference between its starting height and the height to which it is raised.
Kinetic energy is the energy a moving object has by virtue of its movement, and is equal to half the mass of the object times the square of its velocity. The appropriate equation is thus:
\begin{equation} E = \frac{mv^2}{2} \end{equation}Potential energy and kinetic energy can be interchanged, although always with some loss (as heat). Thus if a ball is lifted to a certain height, it gains potential energy. If it is then dropped, the potential energy is converted into kinetic energy as it looses height but gains speed. It then hits the ground and compresses like a spring. The kinetic energy is converted back into potential energy in the form of elastic strain energy (but with some loss in the form of heat - dead energy). The ball then expands again, and bounces up, thus converting the potential energy back into kinetic energy. And so on, until all the energy is lost into heat, and the ball stops bouncing.
Power is the rate of delivering energy, i.e. power is energy divided by time. We have seen above that potential energy is force times distance, and we all know that velocity is distance divided by time. So if we put these various equations together we get:
\begin{equation} P = \frac{E}{t} = \frac{f \times d}{t} = f\times v \end{equation}If we consider an expanding spring, then its power output is the force with which the spring pushes, times the velocity of movement of the object it pushes.
We now consider how this applies to grasshoppers jumping
Newton's laws of motion tell us that a grasshopper (or anything else) that jumps a horizontal distance of 1 m must have a take-off velocity of about 3 m s-1.
If the grasshopper has a mass of 0.003 kg (i.e. it weighs 3 g), then its kinetic energy at the moment of take off must be about 13 mJ.
If we assume that acceleration is uniform while the grasshopper pushes against the ground, then the average velocity during the thrust period is half the take-off velocity , i.e. 1.5 m s-1. The distance over which the grasshopper accelerates is the combined length of the femur and tibia, which is about 45 mm. This means that the time from the start of the push to the moment of take-off takes about 30 ms (\(t = d/v\)). So the average power output is 0.43 W (\(P = E/t\)), and the average thrust force is 0.29 N (\(f = P/v\)), which is about 10 times the weight of the animal.
The geometry of the leg and the physiology of the muscles means that the force is most definitely not constant during the acceleration phase, and this complicates things. The measured peak power output during acceleration of a locust weighing 1.7 g with a leg length of 30 mm is in fact about 0.75 W.
As before, most of these measurements were obtained using high-speed photography of actual jumping grasshoppers (see Bennet-Clark, 1975, in Bibliography).
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