A Near Miss! In the early morning hours of June 14,2002, the...

A Near Miss! In the early morning hours of June $14,2002,$ the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated 2002 $\mathrm{MN}$ , remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was $73,600$ miles from the center of the Earth-about a third of the distance to the Moon. (a) Find the speed of the asteroid at closest approach, assuming its speed at infinite distance to be zero and considering only its interaction with the Earth. (b) Observations indicate the asteroid to have a diameter of about 0.730 $\mathrm{km}$ . Estimate the kinetic energy of the asteroid at closest approach, assuming it has an average density of 3.33 $\mathrm{g} / \mathrm{m}^{3} .$ (For comparison, a 1 -megaton nuclear weapon releases about $4.2 \times 10^{15} \mathrm{J}$ of energy.)

A Near Miss! In the early morning hours of June $14,2002,$ the Earth had a remarkably close encounter with an asteroid the size of a small city. The previously unknown asteroid, now designated
2002 $\mathrm{MN}$ , remained undetected until three days after it had passed the Earth. At its closest approach, the asteroid was $73,600$ miles from the center of the Earth-about a third of the distance to the Moon. (a) Find the speed of the asteroid at closest approach, assuming its speed at infinite distance to be zero and considering only its interaction with the Earth. (b) Observations indicate the asteroid to have a diameter of about 0.730 $\mathrm{km}$ . Estimate the kinetic
energy of the asteroid at closest approach, assuming it has an average density of 3.33 $\mathrm{g} / \mathrm{m}^{3} .$ (For comparison, a 1 -megaton nuclear weapon releases about $4.2 \times 10^{15} \mathrm{J}$ of energy.)

Key Concepts

Volume of a Sphere

The volume of a sphere is calculated using the formula (4/3)?r³, where r is the radius of the sphere. This geometric relationship is essential for estimating the mass of spherical objects when their density is known, thereby connecting geometry with physical properties in problems involving astronomical bodies.

Escape Velocity

Escape velocity is the minimum speed an object must have to break free from the gravitational pull of a celestial body without further propulsion. This concept connects kinetic energy to gravitational potential energy and is used in calculations where an object starts from infinity with zero velocity or in reverse, as seen when evaluating speeds due to gravitational attraction.

Density and Mass Estimation

Density, defined as mass per unit volume, provides a way to estimate the mass of an object when its size or volume is known. When combined with geometric formulas (such as the volume of a sphere), this concept allows one to derive the mass from the object's average density, which is particularly useful when dealing with astronomical objects where direct measurement of mass is impractical.

Gravitational Potential Energy

Gravitational potential energy is the energy an object has by virtue of its position in a gravitational field. It is typically given by the formula -GMm/r, where G is the universal gravitational constant, M is the mass of the body (for example, Earth), m is the mass of the object, and r is the distance between their centers. This concept is key to understanding energy changes during gravitational interactions.

Conservation of Energy

This principle states that the total mechanical energy in an isolated system remains constant. In the context of an object falling into or escaping from a gravitational field, the loss in gravitational potential energy is converted into kinetic energy, and vice versa. This concept is essential for determining speeds and energies when only gravitational forces are acting.

Kinetic Energy

Kinetic energy is the energy an object possesses owing to its motion, defined by the expression (1/2)mv², where m is the mass and v is the velocity of the object. In problems involving gravitational acceleration, converting potential energy into kinetic energy helps determine the speed of an object when it reaches a particular point in its trajectory.

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