It is a fundamental fact of life that ants cannot be as large as elephants, and whales cannot be as small as goldfish. For every type of animal there is a most convenient size, Each animal has a characteristic length L. From this length one can define a characteristic area which is proportional to L 2 , and a characteristic volume which is proportional to L 3 . The characteristic mass is also taken to be proportional to L 3 .

The maximum size of an animal is constrained by the material out of which it is made. For example, consider whether a giant human is a viable creature. If we rescaled a human by a factor of 10, we would necessarily increase the radius of the legs by a factor of 10. However, the mass of the human would increase by a factor L 3 , or 1000. Therefore, legs with 100 times the cross-sectional area would have to support 1000 times the weight.

Size also explains why an ant that off the roof of a four-story building will not be seriously injured, whereas a human most certainly would be injured. A small animal can survive such a fall because of its relatively large surface area to volume ratio. An object in free fall within the Earth’s atmosphere will only accelerate up to a certain velocity because of friction caused by the air resistance. The frictional force can often be approximated as being proportional to the velocity times the cross-sectional area of the falling object. The maximum velocity that an object reaches in free-fall is called the terminal velocity.

(i) Which of the following statements must be assumed if the characteristic mass of an animal is taken to be proportional to L 3 –

(I) Every animal can be assigned a characteristic length L.

(II) The characteristic volume of the animal is proportional to L 3 .

(III) The density of the animal is constant throughout that particular animal.