A matter of scale: roughly speaking, grasshoppers (scale size 2 cm) and people (

Question

A matter of scale: roughly speaking, grasshoppers (scale size 2 cm) and people (scale size 2 m) are capable of jumping to about the same height, say 1/2 meter (we shall ignore the wings and flying capabilities of grasshoppers). Suppose there exists a monster grasshopper 100 times larger than normal, or about the same size as a person. How high do you think such a creature might be capable of jumping? Assume the monster hopper has the same basic muscle strength and other relevant physiological factors as ordinary grasshoppers and, indeed, people. Do you think it will jump 100 times higher than people and ordinary grasshoppers? Ten times higher? Roughly the same? Back up your answer with scaling arguments.

Note that the height H that any physical object, such as batteries, grasshoppers and people can jump or climb powered by the energy stored within the object is a measure of the “specific energy”, the available stored energy U per unit mass m of the object, H = (U/m)/g (This result is consistent with the scaling law H v2/g ) One of the challenges in designing electric automobiles, for example, is to develop batteries with greater specific energy so that the car can travel further between charges. Can you estimate the relative specific energy of an ordinary AA battery compared with the same volume of gasoline? Their ratio will be comparable to the ratio of the height that battery operated devices can climb compared to gasoline powered climbers. The energy stored in a battery is the product of the voltage times the electric charge stored which, in turn, is related to the number of Ampere-hours, the usual figure-of-merit describing battery capacity.

Explanation / Answer

As H = (U/m)/g Hence as the gross hopper is 100 times larger it's mass also increases by 100 times. And as H is inversly proportional to m Hence the height decreases by 100 times or (0.5)/100 = 0.005 m only .

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