Suppose you can jump 55 cm high from a standing start on Earth. Find the radius of the largest

step 1 In this problem, we have to find the radius of the largest spherical asteroid. Let this radius i

Suppose you can jump 55 cm high from a standing start on...

Suppose you can jump $55 \mathrm{~cm}$ high from a standing start on Earth. Find the radius of the largest spherical asteroid you could escape by jumping, assuming a uniform asteroid density of $4000 \mathrm{~kg} / \mathrm{m}^{3}$

Key Concepts

Energy Equivalence for Escape Conditions

This concept involves setting the kinetic energy that an individual can generate (as evidenced by a jump) equal to the gravitational energy barrier required to escape a celestial body’s gravitational field. By equating these energies, one can derive a condition that determines the maximum size (in terms of radius) of a celestial body from which escape is possible.

Gravitational Field of a Sphere

For a spherical object with symmetric mass distribution, the gravitational field outside the sphere can be treated as if all the mass were concentrated at its center. This principle allows the use of standard formulas for gravitational potential energy and escape velocity, directly relating the mass and radius of the sphere.

Uniform Density Model

Assuming a uniform density simplifies the calculation of a celestial body’s mass by using the formula for the volume of a sphere multiplied by the density. This approach reduces the complexity of gravitational calculations by ensuring that the mass distribution is consistent throughout the asteroid.

Escape Velocity

Escape velocity is the minimum speed required for an object to break free from the gravitational pull of a celestial body without further energy input. It is derived by equating the kinetic energy needed to overcome the gravitational potential energy, and its formula is crucial for determining whether a given energy (from a jump, for example) is sufficient to escape a gravitational field.

Kinematic Energy Conversion

This concept explains how kinetic energy is converted into gravitational potential energy during a jump. By using the energy conservation principle, one can relate the initial kinetic energy (obtained from jumping) to the height reached in a gravitational field through the equation mgh, where m is mass, g is the acceleration due to gravity, and h is the height.

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