Problem 47 - College Physics

Question

\(\bullet\) \(\bullet\) Human energy vs. insect energy. For its size, the com- mon flea is one of the most accomplished jumpers in the animal world. A \(2.0-\mathrm{mm}\) -long, 0.50 \(\mathrm{mg}\) critter can reach a height of 20 \(\mathrm{cm}\) in a single leap. (a) Neglecting air drag, what is the take- off 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 kg, 2.0 -m-tall human could jump to the same height com- pared with his length as the flea jumps compared with its length, how high could he 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?

Step-by-Step Solution

Answer

(a) 2.0 m/s, (b) 1.0 µJ and 2.0 J/kg, (c) 200 m jump, 62.6 m/s speed, (d) 5.88 J/kg, (e) energy is stored in resin pads.

Step 1: Understand the Problem

We need to find the takeoff speed of a flea jumping 20 cm ( (a) ), calculate its kinetic energy at takeoff, and per kilogram of mass ( (b) ). For a human, compare jumping heights relative to body length and find required takeoff speed ( (c) ). Lastly, compute the kinetic energy per weight for a 65kg human jumping 60 cm ( (d) ) and locate where the flea stores its leap energy.

Step 2: Calculate Flea's Takeoff Speed

Using the energy conservation principle for part (a), we equate the kinetic energy at takeoff to the potential energy at the peak height:\[ \frac{1}{2} m v^2 = mgh \]Where- \(v\) is takeoff speed,- \(h = 0.2\, m\) is the height,- \(g = 9.8\, m/s^2 \) is the acceleration due to gravity.Cancelling mass \((m)\) and solving for \(v\):\[ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 0.2} \approx 2.0\, m/s \]

Step 3: Calculate Flea's Kinetic Energy

The kinetic energy \(KE\) at takeoff is given by:\[ KE = \frac{1}{2} mv^2 \]with \(m = 0.5\,mg = 0.5 \times 10^{-6} kg\) and \(v = 2.0\, m/s\):\[ KE = \frac{1}{2} \times 0.5 \times 10^{-6} \times (2.0)^2 = 1.0 \times 10^{-6} J\]The kinetic energy per kilogram is:\[ \frac{KE}{m} = \frac{1.0 \times 10^{-6}}{0.5 \times 10^{-6}} = 2.0\, J/kg\]

Step 4: Human Jump Height and Speed

Relative jump height compared to the length for a flea is \(\frac{0.2 m}{0.002 m} = 100\). For a human of length 2 m:\[ \text{Jump Height} = 2 m \times 100 = 200 m\]For takeoff speed (using the same energy equation):\[ v = \sqrt{2gh} = \sqrt{2 \times 9.8 \times 200} \approx 62.6\, m/s\]

Step 5: Human Kinetic Energy for 60 cm Jump

For a 65 kg human jumping 60 cm:\[ KE = mgh = 65 \times 9.8 \times 0.6 \approx 382.2 J\]Kinetic energy per kilogram:\[ \frac{KE}{m} = \frac{382.2}{65} \approx 5.88\, J/kg\]

Step 6: Flea Energy Storage

The flea stores its energy in a specially adapted pad of a protein called resilin, located in its hind legs, which allows it to store elastic potential energy and release it rapidly during a jump.

Key Concepts

Energy Conservation

Energy conservation is a fundamental principle in physics. It states that the total energy in an isolated system remains constant over time. For the jumping flea, energy conservation helps us understand how energy transforms from one type to another. When the flea leaps, its kinetic energy—the energy of motion—is initially at its maximum. As it rises, this kinetic energy converts into potential energy—the energy of position—due to gravity.

At its peak jump height, the flea's kinetic energy is fully converted into gravitational potential energy. Then, as it descends, the potential energy converts back into kinetic energy. By using the energy conservation formula, \[\frac{1}{2} m v^2 = mgh\]where \(m\) is mass, \(v\) is takeoff speed, \(h\) is height, and \(g\) is the acceleration due to gravity, we equate the kinetic energy at takeoff to potential energy at the jump height.

Potential Energy

Potential energy is the stored energy of an object due to its position or state. In the context of the jump, it primarily refers to gravitational potential energy, which depends on the height above the ground. The flea converts this stored energy back into kinetic energy to reach its leap height of 20 cm.

When we look at potential energy, the equation \(PE = mgh\) lets us calculate how much energy is stored based on the flea's mass \(m\), the height \(h\) it reaches, and the gravitational constant \(g = 9.8 \text{ m/s}^2\). For more complex organisms like humans, potential energy analysis helps understand how energy assists in larger jumps when considering mass and height difference.

Jump Mechanics

Jump mechanics involve studying how forces and body movements facilitate the leap. For both fleas and humans, the mechanics include converting stored energy into upward motion. Factors such as muscle strength, elasticity of body parts, and body alignment play crucial roles.

Fleas achieve impressive leaps due to their unique anatomy. The power is primarily generated by their hind legs, which work together like a catapult. In humans, jumping mechanics vary, emphasizing a powerful push-off from the ground using leg muscles. When analyzing jump mechanics in humans relative to their body size, a comparative model can show that jump height influences required takeoff speed.

Elastic Potential Energy

Elastic potential energy is a type of potential energy stored when an object is deformed, such as being stretched or compressed. The flea remarkably uses its elastic potential energy during jumping. Inside its legs, a natural 'spring' mechanism made from resilin—a highly elastic protein—stores energy.

This energy is gradually accumulated when the flea readies itself to spring. When released, the stored elastic potential energy converts quickly into kinetic energy, enabling the incredible jumps. Understanding this biomechanics concept provides insight into how living organisms exploit elastic structures to enhance movement efficiency and power.