For its size, the common flea is one of the most...

For its size, the common flea is one of the most accomplished jumpers in the anima world. A 1.5 -mm-long, $0.55 \mathrm{mg}$ flea can reach a height of $17 \mathrm{~cm}$ it a single leap. (a) Ignoring air drag, what is the takeoff speed of sucl a flea? (b) Calculate the kinetic energy of this flea at takeoff and it: kinetic energy per kilogram of mass. (c) If a $72 \mathrm{~kg}, 1.70-\mathrm{m}$ -tall humar could jump to the same height compared with his length as the fle jumps compared with its length, how high could the human jump, ang what takeoff speed would the man need? (d) 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 $72 \mathrm{~kg}$ person? (e) Wher does the flea store the energy that allows it to make sudden leaps?

For its size, the common flea is one of the most accomplished jumpers in the anima world. A 1.5 -mm-long, $0.55 \mathrm{mg}$ flea can reach a height of $17 \mathrm{~cm}$ it a single leap.
(a) Ignoring air drag, what is the takeoff speed of sucl a flea? (b) Calculate the kinetic energy of this flea at takeoff and it:
kinetic energy per kilogram of mass.
(c) If a $72 \mathrm{~kg}, 1.70-\mathrm{m}$ -tall humar could jump to the same height compared with his length as the fle jumps compared with its length, how high could the human jump, ang what takeoff speed would the man need?
(d) 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 $72 \mathrm{~kg}$ person? (e) Wher does the flea store the energy that allows it to make sudden leaps?

Key Concepts

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion and is given by the formula ½mv², where m is the mass and v is the velocity. In jumping mechanics, the kinetic energy at takeoff determines the work done to raise the body against gravity, and it can be used to compute the required takeoff speed given the mass and the height achieved during the jump.

Gravitational Potential Energy

Gravitational potential energy represents the energy an object has due to its position in a gravitational field and is typically calculated by mgh, where m is mass, g is the acceleration due to gravity, and h is the height. In problems concerning jumps, the energy gained in lifting the body to a certain height is derived from the conversion of kinetic energy into gravitational potential energy.

Scaling and Proportions in Biology

Scaling laws help understand how different sized organisms perform similar actions, such as jumping. When comparing the performance of small creatures to larger ones, proportional relationships between body dimensions and strength or power output are considered. This concept is crucial when determining how a characteristic like jump height or takeoff speed scales with the size or mass of the organism.

Elastic Energy Storage in Biological Systems

Many animals rely on elastic structures to efficiently store and rapidly release energy during movements such as jumping. In biological systems, proteins and specialized tendons can act as springs, allowing energy to be accumulated over time and then quickly discharged. This mechanism is especially important for organisms that require sudden, high-powered movements, such as in high jumps or rapid accelerations.

Conservation of Energy

This concept states that energy cannot be created or destroyed, only transformed from one form to another. In the context of jumping, the kinetic energy produced at takeoff is converted into gravitational potential energy at the peak of the jump. Ignoring losses such as air drag, the energy imparted during the jump is all converted into the energy needed to overcome gravity, which allows us to relate the takeoff speed to the maximum height reached.

Transcript

00:01 Hello, my name is andrew and i'll be solving this problem from the university physics with modern physics textbook by hugh young and roger friedman.

00:13 So this problem has five different parts.

00:17 So i'll start with part a.

00:20 So in part a, the question is asking, ignoring air drag, what is the takeoff speed of such a fleet? and so for this problem, what you would need to do is you would use the velocity formula provided.

00:36 So in this case, it's v1 is equal to two times the gravity times the height.

00:45 And so we are given the height that the fleet is able to travel, and it is 17 centimeters, but we have to convert this to meters so we can have it in si units.

00:55 So once we do that, we can input the variables in our equation for velocity.

01:05 So we'll have the square root of two times 9 .80 meters per second squared times 0 .17 meters.

01:16 And we're going to end up with 1 .83 meters per second, and this is the units of velocity, so this is correct.

01:25 It is important to note that some ways of solving this problem is to round the 0 .17 meters to 0 .2 meters or 0 .20 meters.

01:38 And by doing that, you'll get two meters per second.

01:44 It's just depending on which way you're trying to round to make the calculations easier.

01:50 I decided to go ahead and leave it as 0 .17 meters.

01:55 Part b wants us to calculate the kinetic energy per kilogram.

02:00 So we can go ahead and use the equation for conservation of energy.

02:05 And so we have k1 plus u1 is equal to k2 plus u2, and we know that u1 is equal to zero because there is no potential energy at the start since it's moving.

02:16 And then when it reaches its max height, it's going to be, for a brief moment, it's going to be stopped.

02:24 So then the kinetic energy at that point will be zero, and it'll have the greatest potential energy since it's at its max height.

02:33 And so we can go ahead and then have k1 is equal to u2, and u2 is just equal to mgh.

02:40 That's the potential energy.

02:41 From here, what we can go ahead and do is we can go ahead and plug in the values that we are given.

02:47 So for the mass it's given to us in milligrams, and we need to convert it to kilograms so we can have it in the units.

02:58 So this is just, we can do that by multiplying times 10 to the negative six, and this will convert it for us.

03:05 And then we have gravity, 9 .80 meters per second squared, and then we have the 0 .17 meters.

03:10 So we can go ahead and multiply this, and this is going to give us 9 .2 times 10 to the negative seven joules.

03:17 And in order to get the ratio for kinetic energy and mass, we can just divide by the mass.

03:24 So we're going to go ahead and do that.

03:26 So, and we then are, we obtain a value of 1 .7 joules per kilogram.

03:34 The, since we are dividing by the mass, we did not necessarily need to do that in the first step since we're going to cancel it out.

03:40 But this is just good to show that, you know, when we're multiplying the units kilograms over here, multiplying meters per second squared, and then meters, we're going to go ahead and get joules.

03:54 So it's just a good thing to know that those units are equivalent to joules.

03:59 And so this is just for that sake of that, to make sure our units are consistent...

Please give Ace some feedback