T-Rex Speed: Analyzing the Effects of Gravity

[clearwater304]

I was reading in my fluid mechanics book how the speed of the t-rex was calculated through dimensionless analysis.
They plotted v^2/gl vs s/l
v=velocity
g=acceleration due to gravity
l=leg length
s=stride
Which showed that the data for most animals fell approximatly on one curve. My book states most people could outrun a t-rex. My question is, does gravity have a greater effect on larger animals limiting their velocity. For instance, would a jumbo sized version of myself run faster, slower, or have the same speed?

http://en.wikipedia.org/wiki/Tyrannosaurus

 

[e.bar.goum]

My question is, does gravity have a greater effect on larger animals limiting their velocity.

Well, no, because the acceleration of a body due to gravity is independent of mass. Think Galileo etc.

But in any case, I don't think that answers the question of whether a jumbo sized version of yourself would run faster. That rather depends on how other factors (biological factors, like how fast you can move your legs as a function of their mass) scale with size.

 

[clearwater304]

I'm thinking a small change in leg length would cuase a dramatic change in mass. I'm guessing this would require more energy to move once the object is above a peak mass, which has more to do with biomechanics. Thats why I was thinking gravity would play a role, as far as the energy required to lift an object, but then again, it's not directly implied by the above equation. I'm still not sure, I think this article states larger animals move quicker (page 13)

http://jeb.biologists.org/content/185/1/71.full.pdf

 

[Drakkith]

As the article on wikipedia reads, there are MANY different views on how fast a T-rex could run, ranging from a complete inability to run to being able to run over 40 mph. The reason for the wide range is because there are many different variables.

For a bigger version of yourself, I don't think it's possible to scale a human being to more than a certain size or we start running into problems. I'm not sure though, I think it's something I read once, I just can't remember where.

 

[JHamm]

Having a longer stride will help you run faster (think the difference between a running ant and you) but you can't just "upscale" a body and expect it to work the same.

 

[Ryan_m_b]

A bigger human body would have a longer stride but requires more energy to move, generates higher wind resistance and would suffer from medical problems if too big e.g knee problems, heart problems etc

 

[Andy Resnick]

I was reading in my fluid mechanics book how the speed of the t-rex was calculated through dimensionless analysis.
They plotted v^2/gl vs s/l
v=velocity
g=acceleration due to gravity
l=leg length
s=stride
Which showed that the data for most animals fell approximatly on one curve. My book states most people could outrun a t-rex. My question is, does gravity have a greater effect on larger animals limiting their velocity. For instance, would a jumbo sized version of myself run faster, slower, or have the same speed?

Clever (dimensional) analysis: the ratio of centripetal force and gravitational force plotted vs. the scaled stride.

http://www.engin.umich.edu/class/me646/Lecture%2012%20Step%20length%20vs%20speed.pdf

The dimensionless velocity above is also the Froude number (2/3 of the way down):

http://en.wikipedia.org/wiki/Froude_number

http://www.sciencedirect.com/science/article/pii/S0966636204000268

 

[clearwater304]

So I guess the fact that a larger animal would have a longer stride would give it more speed, but if we consider Kleiber's law, you would have to take into account that smaller animals have a faster metabolism, thus more energy readily available. I would think there would be some sort of optimum speed vs size ratio based on this.

http://universe-review.ca/R10-35-metabolic.htm

 

[tahayassen]

Wait. Why are you guys saying that you run into problems when you upscale a human body and keep all the proportions the same? Is it because you'll get different gravitational pulls? Gravity will pull harder on your feet than your head. But doesn't that apply to the small version too?

 

[Drakkith]

Wait. Why are you guys saying that you run into problems when you upscale a human body and keep all the proportions the same? Is it because you'll get different gravitational pulls? Gravity will pull harder on your feet than your head. But doesn't that apply to the small version too?

I think it's due more to the fact that as you scale something up its volume increases faster than its surface area.

See here: https://www.physicsforums.com/showthread.php?p=2893766

 

[Ryan_m_b]

Wait. Why are you guys saying that you run into problems when you upscale a human body and keep all the proportions the same? Is it because you'll get different gravitational pulls? Gravity will pull harder on your feet than your head. But doesn't that apply to the small version too?

I think it's due more to the fact that as you scale something up its volume increases faster than its surface area.

See here: https://www.physicsforums.com/showthread.php?p=2893766

This is exactly right although there are other problems as well such as the tolerance of muscles/cartilage/bone, the strength of the heart muscle etc.

 

[A.T.]

I think it's due more to the fact that as you scale something up its volume increases faster than its surface area.

Yes and the force of muscles depends on their cross sectional area, not volume. So when you scale a human by factor 2, his muscles are 4 times stronger, but he is are 8 times heavier.

Based on this simplistic approach you can state, that if it was possible to scale a human down to the size of an ant, he could carry much more than a ant can. But there are many assumptions involved here.

 

[Andy Resnick]

Wait. Why are you guys saying that you run into problems when you upscale a human body and keep all the proportions the same? Is it because you'll get different gravitational pulls? Gravity will pull harder on your feet than your head. But doesn't that apply to the small version too?

Not just a human body- anything. Two bridges, each identical in every ways except for a (fixed) change in length scale, can carry (proportionally) different loads. This is a general rule when one property (say mass) varies as the volume by another property (say strength) varies as a (typically cross-section) area. Animals with large heads (bulls, elephants) have necks that are considerably different in form than smaller animals, even though the overall body proportions may be similar.

A second factor: Different forces scale differently with length: gravity with volume, surface tension with area. Insects can live on the surface of water, while larger animals cannot- and their anatomy has evolved in response to this. Insects don't have lungs- their tissues can be supplied by oxygen by diffusion directly- they don't need lungs or a circulatory system.

As for hydrostatic pressure heads, that definitely is important for a giraffe! Consider the pressure change experienced by the cranial capillaries when the animal lowers its head to drink.

 

[Richy-Rich]

You would generate the same amount of energy and speed but since your legs would be longer, then you would cover more distance, thus making you "FASTER".

 

[Ryan_m_b]

You would generate the same amount of energy and speed but since your legs would be longer, then you would cover more distance, thus making you "FASTER".

Beyond a certain extent this is not true. A larger body has more mass but not the muscle strength to compensate for it. A smaller person with a lighter body can cover more distance with faster, shorter strides than a heavier person with slower, longer strides. Of course there are many other factors.