BIO Human Energy vs. Insect Energy. For its size, the common...

BIO Human Energy vs. Insect Energy. For its size, the common flea is one of the most accomplished jumpers in the animal world. $A 2.0$ -mm-long, 0.50 -mg critter can reach a height of 20 $\mathrm{cm}$ in a single leap. (a) Neglecting air drag, what is the takeoff speed of such a flea? (b) Calculate the kinetic energy of this flea at takeoff and its kinetic energy per kilogram of mass. (c) If a $65-\mathrm{kg}, 2.0-\mathrm{m}-$ tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could the human jump, and what takeoff speed would he need? (d) In fact, most humans can jump no more than 60 $\mathrm{cm}$ from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 65-kg person? (e) Where does the flea store the energy that allows it to make such a sudden leap?

Key Concepts

Energy per Unit Mass

This concept refers to the amount of energy available or required per kilogram of an organism’s mass. Measuring kinetic energy on a per kilogram basis facilitates comparisons of performance across species of vastly different sizes by normalizing energy outputs, thereby highlighting the efficiency and power of small organisms like fleas relative to larger ones.

Scaling Laws and Size Comparisons

Scaling laws in biomechanics address how physical and physiological parameters change with the size of an organism. When comparing the jump capabilities of a flea and a human, these laws help explain how relative performance (e.g., jump height relative to body size) can vary greatly despite differences in absolute energy output, due to differences in mass, muscle power, and biomechanical design.

Elastic Energy Storage in Biological Structures

Many animals, particularly small jumping insects like fleas, utilize elastic energy storage mechanisms. Instead of relying solely on muscle contraction, these organisms store energy in elastic structures such as modified exoskeletons or specialized connective tissues. When this stored energy is rapidly released, it significantly amplifies the power output, enabling extraordinary feats like the flea’s high jump relative to its size.

Conservation of Energy

This concept involves the principle that energy cannot be created or destroyed, only converted from one form to another. In the context of jumping, a creature’s kinetic energy at takeoff is entirely converted into gravitational potential energy at the peak of the jump, assuming negligible air resistance. This relationship allows us to relate the jump height to the initial speed and energy output regardless of the scale of the jumping organism.

Kinetic Energy

Kinetic energy is the energy associated with an object’s motion and is calculated by the formula ½mv², where m is the mass and v is the velocity. In jumping analyses, this concept is used to determine how much energy is needed at takeoff to achieve a particular jump height, and it allows comparison across organisms by also considering energy per unit mass.

Transcript

00:01 Problem with five different parts, we'll address them one at a time, but it all has to do more or less with a flea or a small insect or bug that is two millimeters tall it is or long has a mass of 0 .05 milligrams and can jump up to a height of 20 centimeters.

00:22 So this thing can jump upwards to a height of 20 centimeters so the first excuse me the first part is actually asking us to solve for the final velocity of the bug.

00:34 Well, to solve for part a, we've got the potential energy due to gravity at the end of the jump.

00:41 So if they're jumping up 20 centimeters, going up 20 centimeters, that potential energy at the end is equal to the kinetic energy beginning.

00:50 This gives us mgh, potential energy due to gravity, is equal to one half, mass.

01:02 Velocity squared of kinetic energy.

01:05 All right, mass cancels out.

01:09 Right away we get v is equal to 2gh.

01:15 And our h this time, our h is over here.

01:19 Our h is 20 centimeters, so 0 .2 meters.

01:26 And we take the whole square root of that, because we've got on square both sides, get v squared.

01:31 And if you solve that with calculator you get just about 2 1 .98 meters per second great so that's part a this is part a part b we're asked to figure out what is the energy density what is or no the kinetic energy per kilogram so yes sort of energy density what is a per kilogram of mass.

02:09 Let me delete this.

02:10 So we're going to use the same idea that potential energy equals the kinetic energy.

02:18 But this time we have, we're trying to solve for just the kinetic energy over the mass.

02:26 Well, if we substitute kinetic energy equals potential energy of gravity, again, we'll get m -g -h over m.

02:35 Well, again, mass is cancel.

02:38 Masses cancel and since we just see that this answer is just g times m and g 9 .8 meters per second square the acceleration due to gravity times the height that it jumps up to again 0 .2 meters you plug that into your calculator and for part b we get a kinetic energy per kilogram of 1 .96 joules per kilogram all right.

03:13 So that is part b.

03:18 Part c now asks how tall or how high would a human have to jump if they jump to the same height that a flea jumps to compared to its own height? well, a flea is 2 millimeters tall, so the flea, i can get rid of this too for now, the flea is 2 millimeters tall, jumps to 20 centimeters.

03:49 That is a difference of 100, so 20 centimeters divided by 0 .2 millimeters, so 0 .2 millimeters.

03:59 So 2 millimeters is 0 .2 millimeters.

04:06 I just did the conversion.

04:08 Excuse me.

04:09 So this is converted now to centimeters.

04:12 Center is canceled, we get a factor of 100, right? okay, so that just means that the flea has a flea jumped 100 times its height.

04:20 Now, how tall is the person to jump if they jump 100 times their height? well, they're going to have to jump the person.

04:28 So the height that they have to jump to is now going to be 200 meters...