Group Members: Burt Kessler, Harold Meiselman, Cynthia Miyada, Soo Boo Tan
Research Question: Does Alexander’s formula for dimensionless speed apply to a hopping motion such as that of the cricket (Acheta domesticus) or the frog (Litoria caerulea)?
Hypothesis: The Dimensionless Speed for the hopping motion of a cricket or frog will exceed the projected value based on Alexander’s formula.
Background: R.McNeill
Alexander made extensive observations and analyses of the biomechanics
of animals in motion. This led him to apply a principle first used by William
Froude (a naval engineer using small-scale models to test new ship designs)
to the locomotion of animals. He suggested that the Froude number, equal
to
(speed of locomotion)2
gravitational acceleration x leg length
could be used to compare the locomotion of animals of different sizes
with regard to speed and relative stride length (stride length divided
by leg length) independent of the dimensions of the animal. This is referred
to as dimensionless speed. He demonstrated (using of a graph of
relative stride length plotted against the square root of the Froude number)
that a wide range of mammals of different sizes have about the same
relative stride lengths when traveling with equal Froude numbers.
Acheta domesticus (cricket) and Litoria caerulea (White's tree frog) were selected as representatives of an invertebrate and a vertebrate because they were readily available, easy to care for, and used hopping as a primary mode of locomotion.
Materials
A few small crickets
A White’s Dumpy tree frog
Measuring tape
Stop watch
White paper roll
Marker pens
Experimental Design:
1. Measure length of cricket’s hind limb (femur, tibia/fibula and ankle)
to the nearest centimeter.
2. Place ten feet by three feet of white paper towel on flat area (floor).
3. Place cricket on paper.
4. Start the stopwatch.
5. With marker pen, mark each spot the cricket lands on.
6. Repeat until cricket jumps out of paper range.
7. Stop the stop watch at the last landing before the cricket leaves
the paper.
8. Record the total time taken for the jumps in seconds.
9. Measure the lengths to the nearest millimeter of each of the
jumps using the measuring tape.
10. Repeat the above procedure at least 3 times for each cricket.
11. Repeat for at least 3 subjects.
12. Repeat steps 1 ? 10 with frog at least 10 times
Data: available on request
Possible Sources of Experimental Error or Inaccuracy:
1.A wristwatch with stopwatch functions was used due to the lack of
access to an actual stopwatch, thereby increasing potential for measurement
error.
2. Brief and inconsistent pauses between jumps were included in the
time measurements used to calculate speed.
3. Efforts were not taken to ensure that subject organisms were in
optimum health and physical condition.
4. Subsequent background reading revealed that Alexander had applied
this concept of dimensionless speed only to the comparison of walking and
running in mammals because their structures are geometrically similar and
therefore their movements are dynamically similar. These observations were
not intended to be applicable to widely diverse types of organisms with
differing modes of locomotion.
Questions and/or Suggestions for Future Research
1.Design and perform experiments to test and to extend
Alexander's work relating to hopping/jumping type movements.
2. Further investigate the relationship between speed, leg length,
hopping height and distance and energy expenditure in jumping animals.
3. Describe and analyze the biomechanics of hopping or jumping motions.
Address the roles of levers and projectile trajectory in this mode of locomotion.
4. How do anatomical and / or physiological differences between vertebrates
and invertebrates account for the widely differing observations and data
relating to the above considerations?
Conclusions
The data collected supported our hypothesis that hopping motions such
as that of frogs and crickets does not conform to Alexander’s curve for
relative stride over dimensionless speed (square root of the Froude number).
Like Alexander’s curve, there is a positive correlation of speed with stride/jump
length. The greater the speed, the longer the jump. Because
the slope is so slight, this relationship is weak.
Subsequent reading from Alexander’s ExploringBiomechanics
indicates that there exists a relationship between the speed and angle
at which the animal takes off and the distance it covers in hopping or
leaping motions. This is described by the formula:
mgh = 1/2 mv2
This involves measurement of kinetic & potential energies at take-off
and height of jump.
Bibliography
Alexander, R. McNeill. Exploring Biomechanics.1992: Scientific
American Library