The maximum energy a bone can absorb without breaking is surprisingly small. Experimental data show that a leg bone of a healthy, 60 kg human can absorb about 200 J. From what maximum height could a 60 kg person jump and land rigidly upright on both feet without breaking his legs? Assume that all energy is absorbed by the leg bones in a rigid landing.

검증된 단계별 안내

Step 1: Identify the given values in the problem. The mass of the person is 60 kg, and the maximum energy the leg bones can absorb is 200 J. The goal is to find the maximum height from which the person can jump without breaking their legs.

Step 2: Recall the formula for gravitational potential energy, which is given by:

E = m g h

, where

E

is the energy,

m

is the mass,

g

is the acceleration due to gravity (approximately 9.8 m/s²), and

h

is the height.

Step 3: Rearrange the formula to solve for height

h

. The equation becomes:

h = \frac{E}{m}

Step 4: Substitute the known values into the formula. Use

E = 200

m = 60

, and

g = 9.8

. The equation becomes:

h = \frac{200}{60}

Step 5: Perform the division and multiplication to calculate the height. This will give the maximum height from which the person can jump without breaking their legs. Ensure units are consistent throughout the calculation.

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주요 개념

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Gravitational Potential Energy

Gravitational potential energy (GPE) is the energy an object possesses due to its position in a gravitational field. It is calculated using the formula GPE = mgh, where m is mass, g is the acceleration due to gravity (approximately 9.81 m/s² on Earth), and h is the height above the ground. In this context, it helps determine how high a person can jump before the energy from the fall exceeds the bone's energy absorption limit.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this scenario, the kinetic energy of the person just before landing is converted into gravitational potential energy during the jump. Understanding this principle is crucial for calculating the maximum height a person can jump without exceeding the energy absorption capacity of their bones.

Rigid Landing

A rigid landing refers to landing in a way that does not allow for any bending or flexing of the legs, which would otherwise help absorb some of the impact energy. In this case, the entire energy from the fall is transferred to the leg bones. This concept is important because it simplifies the analysis by assuming that all the energy from the jump is absorbed by the bones, leading to a direct calculation of the maximum height based on the energy absorption limit.