Energy and Power

Energy

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 appropriate equation is Energy = Force * Distance. 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 thus is Energy = 0.5*Mass*Velocity2

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

Power is the rate of delivering energy, i.e. Power = Energy/Time. We have seen above that Potential energy = Force * Distance, and we all know that Velocity = Distance/Time. So if we put these various equations together we get:
Power = Energy/Time=Force*Distance/Time = Force*Velocity.
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

 

Take-off energy

A grasshopper (or anything else) with a body of mass M moving at a velocity of v has a certain amount of kinetic energy E according to the standard equation for kinetic energy:
E = Mv2/2
This energy is imparted to the grasshopper by the legs thrusting on the ground. The amount of energy depends upon the force F and the distance d over which this force is delivered, i.e. the length of the legs.
E = Fd
A typical grasshopper has about 10 mJ of kinetic energy at the moment of take-off.

 

Take-off power

Acceleration requires that a force be applied continously to an object so that its velocity continuously increases, i.e. it accelerates. This means that the structures which provide the force have to be able to move at high velocity, at least towards the end of the acceleration period. The physical unit which relates force and velocity is power P.
P = Fv
or, since v = d/t,
P = E/t
If the force were applied uniformly throughout leg extension, then the peak power (at the moment of take-off when v is greatest) would be about 0.5 W. In fact, the peak power output is greater than this, at about 0.75 W. 

How were the real data obtained?

As before, most of these measurements were obtained using high-speed photography of actual jumping grasshoppers (see Bennet-Clark, 1975, in Bibliography).