Physics and Grasshopper Legs

Physics and Grasshopper Legs

I had a remarkable experience today which has sent me researching grasshoppers and physics.  As I was leaving the parking lot of my work, I noticed a bright green grasshopper resting on the hood of my car. I didn’t pay that much attention to it, figuring it would get blown off as my car sped up. However, as I accelerated to get onto the interstate, I noticed that the little guy was still there.  As I reached highway speeds of 70mph he was still there. At this point I got curious and decided to perform a little observational science in my car on the way home from work. This post will cover the observations I made and the math I used.  Future posts will attempt to research the biology behind my observations.

You may not have noticed it but I already formulated a hypothesis. My hypothesis was that the grasshopper would be blown off my car when I reached highway speeds. When this hypothesis was proved incorrect by observational data, I attempted to determine at what speed he would be dislodged. This was difficult as I was traveling a highway at rush hour, but I was able to exceed 85mph for 25-30 seconds and the grasshopper remained firmly planted to the hood. This led me to ask how much force that grasshopper had to withstand to remain planted to my hood.

I’ll be honest, such questions are more physics than biology and are thus a bit out of my league. However, I have several friends who are engineers and I asked them for help. One, who I’ll refer to as T. since I don’t have permission to use his name, was very helpful and pointed me to the drag equation used for fluids. In physics air is considered a fluid.  The equation is below.

Drag equation

FD is the drag force, the stylized “p” is the density of the air, the “u” is the velocity, the “A” is the area and the CD is the drag coefficient. I would need to derive all of the numbers in the equation.  The first thing I did was convert from miles per hour into meters per second.  I did this calculation twice, once for maximum speed and once for average speed.  I arrived at the following results:

70mph=31.2928m/s

85mph=37.9984m/s

I then calculated the area of a typical grasshopper using data from a study from the University of Nebraska-Lincoln (link below). By adding the area for the head and the pronotum I determined that:

A=.0280m2

The next two were trickier.  I worked on the stylized “p” next.  For this I had to determine two air pressures, one of dry air, the other of humid air using the following equation:

Pressure of humid air

The stylized “p”s stood for pressure, the “R”s for the gas constants, the “T”s for the temperature in Kelvin. The subscripted “d”s stood for dry air, while the subscripted “v”s stood for humid air. The dry air gas constant is a known value(287.05 J/kgK) and was substituted into the equation. The temperature in Fahrenheit (80) was converted to Celsius using the standard equation and then converted to Kelvin using the standard equation for that resulting in a value of 299.81K.  The stylized “p” of dry air was then calculated using the standard equation for air pressure and was determined to be 98462.08 Pascals.  Therefore, by simple substitution, the pressure of dry air was equal to 1.1466 kg/m

Having determined half of the equation, the more difficult half was yet to be found.  The “T” value was the same but now both the “R” and the stylized “p” values had to be derived. Calculating the density of humid air required the following equation:

humid air 

Es stood for the saturation point of water vapor or hPa. Using this equation required finding the dew point to use in the stylized “p” factorial.  With a quick check of weather bureau data, this was determined to be 25 degrees Celsius. Eso is a known constant.  After performing the math, we arrive at the following result.

Es =31.6701

The above number would serve as our stylized “p” value. Determining the gas constant was harder.  However, with some deriving, I was able to determine it to be roughly 0.0164 Kg/m3.  By simple addition, we determined that the pressure of humid air is 1.163Kg/m3.

Having derived that, I was left with the  C number which is usually determined experimentally, but is tied to the Reynolds number.  Since I had no means to perform an experiment to determine the  CD  number, I derived the Reynolds number and used it for my calculations.  The Reynolds number is determined as follows:

reynolds number

Re is the Reynolds number, the stylized “p” is the density of the fluid, the “l” is something called a chord, which is the length of the object, the “v” is the velocity of the fluid, the stylized “v” is the kinematic viscosity of the fluid and the stylized “u” is the dynamic viscosity of the fluid. Using the second equation,  I substituted the velocities calculated previously in for the “v”. The “l” I used the University of Nebraska-Lincoln data to determine the length or “l” of the average grasshopper which was .008m.  Fortunately, the stylized “v” for kinematic viscosity at 80 degrees F has already be calculated to be 2.09e-5.  By simple replacement, I determined that the Reynolds numbers were:

70mph=11,978Re

85mph=14,545Re.

Having derived all these numbers, I now substituted them into the original equation.

FD =1/2×1.163×31.29282x11,978x.0280=19,474kg/s  at speeds of 70mph

FD =1/2×1.163×31.29282x14,545x.0280=34,868kg/s at speeds of 85mph.

 

I’ve deliberately left off the units for clarity and ease of reading but that is a lot of force. This grasshopper was able to withstand that amount of force to stay on the hood of my car, which offers no obvious opportunities for grip. I’ll be doing further research into this marvelous design and reporting on it as I am able. Our God is amazing and the more I learn about the world around us, the more blown away I am in awe of the majesty he produced.

I’m interested in scientific dialogue on this. Did I do my math right? Is there a known mechanism by which grasshoppers do this? I’d love your feedback.

 

http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1145&context=tnas

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