If air resistance is ignored, the movement of a jumping animal after it leaves the ground is ballistic - like a ball after it's thrown, or a bullet after it's fired from a gun. All the energy for the jump is given during the acceleration to take-off, and once the animal leaves the ground, no further energy is given to it. This is in contrast to, for instance, a flying bird, which continuously exerts energy by flapping its wings, even after it leaves the ground.
The equations describing the kinetics of ballistic movements are well known - they were first worked out by Isaac Newton in the seventeenth century.
Note, in the following equations the units must be consistent.
Distance is measured in metres (m), time is measured in seconds (s), mass is measured in kilograms (kg), force or weight is measured in Newtons (N). Remember, 1 N is the force that will give a mass of 1 kg an acceleration of 1 m s-2; gravity gives it an acceleration of 9.81 m s-2, therefore 1 kg weight [force] is 9.81 Newtons, and 1 N is about 100 g weight. Energy (or work) is measured in Joules. Power is measured in Watts (Joules per second).
If air resistance is negligible (which, for a big grasshopper like a locust, is almost true), then the horizontal range r that a ballistic projectile travels is related to the take-off angle θ and the velocity v at take-off:
\begin{equation} r = \frac{v^2 \sin (2\theta)}{g} \end{equation}where g is the acceleration due to gravity (9.81 m s-2). To maximize range, therefore, an animal should take off at 45o to the horizontal (sin(2θ) = sin(90°) = 1).
The KEY POINT is that if an animal takes off (or a bullet is fired) at this optimal angle of 45°, then its range is entirely dependent on its take-off velocity, whatever the size or weight of the animal.
A grasshopper (or anything else) jumping a distance of 1 m therefore requires a take-off velocity of just over 3 m-1 (\(v=\sqrt{1\times9.81}\)).
An animal which jumps from a standing start has to accelerate its body to take-off velocity from rest. The average acceleration required to achieve a particular velocity depends on the distance d over which acceleration takes place.
\begin{equation} a = \frac{v^2}{2d} \end{equation}The distance in the equation is the distance over which the muscles can exert force while the feet are still in contact with the ground, and this obviously depends on the length of the legs. Thus small animals with short legs have to accelerate disproportionately faster than large animals to achieve the same take-off velocity (if d is small, a has to be large to get the same v). This is one reason why good jumpers, whether human or insect, have long legs relative to body size.
In the locust, d is the combined length of the tibia and femur, which is approximately 0.04 m. This means that the average acceleration as the animal extends its legs is about 92 m s-2 (nearly 10 g). In fact, acceleration is not constant over the whole of leg extension, but rises to a peak at about half extension, and then falls. The peak acceleration can be as high as 180 m s-2.
The force F needed to produce a particular acceleration a depends on the mass M of the body.
\begin{equation} F= Ma \end{equation}So the greater the rate of acceleration the greater the force required for a given mass. If the force were applied uniformly throughout leg extension, it would take about 0.18 N thrust to accelerate a medium sized (2 g) grasshopper to take-off velocity - i.e. about 9 g thrust per leg. In fact, the force, like the acceleration, starts off quite small at about 0.1 N when the animal first starts to move, rises to a peak of about 0.3 N when the legs are half extended, and then falls as the legs reach full extension and the animal leaves the ground.
If we know the jump range, the weight of a grasshopper and the size of its back legs, then many of the energetic parameters of jumping can be estimated from the equations given above. This gives a reasonable first approximation to the answers, but it does assume that force and hence acceleration are uniform throughout leg extension. Owing to the fairly complex geometry of the leg joints, and the physiological properties of the muscle, the assumption is not strictly valid. More accurate data were obtained by experimental measurements made using high-speed photography of actual jumping grasshoppers (see Bennet-Clark, 1975, in the Bibliograph).
Close