Question

For its size, the common flea is one of the most accomplished jumpers in the animal world. A 2.50-mm-long, 0.450 mg critter can reach a height of 16.0 cm in a single leap.

Calculate the kinetic energy of this flea at takeoff.

Calculate the kinetic energy per kilogram of this flea at takeoff.

If a 65.0 kg, 2.00-m-tall human could jump to the same height compared with his length as the flea jumps compared with its length, how high could he jump?

If a 65.0 kg, 2.00-m-tall human could jump to the same height compared with his length as the flea jumps compared with its length, what takeoff speed would he need?

In fact, most humans can jump no more than 60.0 cm from a crouched start. What is the kinetic energy per kilogram of mass at takeoff for such a 65.0 kg person?

Answer 1

Answer 2

To calculate the kinetic energy of the flea at takeoff, we can use the formula:

Kinetic Energy = (1/2) * mass * velocity^2

Given: Length of the flea (L) = 2.50 mm = 0.0025 m Mass of the flea (m) = 0.450 mg = 0.450 * 10^-6 kg Height reached (h) = 16.0 cm = 0.16 m

First, we need to calculate the velocity of the flea at takeoff using the height reached.

Using the conservation of mechanical energy, we have: Potential Energy at takeoff = Kinetic Energy at takeoff

m * g * h = (1/2) * m * velocity^2

Simplifying and solving for velocity, we get: velocity = sqrt(2 * g * h)

where g is the acceleration due to gravity (approximately 9.8 m/s^2).

Plugging in the values, we have: velocity = sqrt(2 * 9.8 * 0.16) ≈ 2.78 m/s

Now we can calculate the kinetic energy of the flea at takeoff: Kinetic Energy = (1/2) * m * velocity^2 Kinetic Energy = (1/2) * 0.450 * 10^-6 * (2.78)^2 Kinetic Energy ≈ 1.09 * 10^-6 J

To calculate the kinetic energy per kilogram of the flea, we divide the kinetic energy by its mass: Kinetic Energy per kilogram = Kinetic Energy / mass Kinetic Energy per kilogram = (1.09 * 10^-6) / (0.450 * 10^-6) Kinetic Energy per kilogram ≈ 2.42 J/kg

Now, let's calculate how high a 65.0 kg, 2.00 m tall human could jump if they could jump to the same height compared with their length as the flea jumps compared with its length.

Height reached by the human = (Length of human / Length of flea) * Height reached by the flea

Length of human = 2.00 m Length of flea = 0.0025 m Height reached by the flea = 0.16 m

Height reached by the human = (2.00 / 0.0025) * 0.16 Height reached by the human ≈ 128.0 m

So, if the human could jump in the same proportion as the flea, they could reach a height of approximately 128.0 meters.

To calculate the takeoff speed the human would need, we can use the formula for velocity at takeoff:

velocity = sqrt(2 * g * h)

where g is the acceleration due to gravity (approximately 9.8 m/s^2) and h is the height reached (0.16 m).

velocity = sqrt(2 * 9.8 * 0.16) ≈ 2.78 m/s

So, the human would need a takeoff speed of approximately 2.78 m/s to jump to the same height as the flea.

Finally, let's calculate the kinetic energy per kilogram of mass at takeoff for a 65.0 kg person who can jump no more than 60.0 cm.

Height reached by the person = 0.60 m Kinetic Energy per kilogram = (1/2) * mass * velocity^2 Kinetic Energy per kilogram = (1/2) * 65.0 * (2.78)^2 Kinetic Energy per kilogram ≈ 151 J/kg