An average-sized asteroid located 5.0 ×10^7 km from Earth with mass 2.0 ×10^13 kg is detected he

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An average-sized asteroid located 5.0 ×10^7 km from Earth...

An average-sized asteroid located $5.0 \times 10^{7} \mathrm{km}$ from Earth with mass $2.0 \times 10^{13} \mathrm{kg}$ is detected headed directly toward Earth with speed of $2.0 \mathrm{km} / \mathrm{s}$. What will its speed be just before it hits our atmosphere? (You may ignore the size of the asteroid.)

Key Concepts

Kinetic Energy

Kinetic energy is the energy an object has due to its motion, given by the expression K = 1/2 m v^2, where m is the mass of the object and v is its velocity. In scenarios where gravitational potential energy is converted to kinetic energy, calculating the change in kinetic energy helps determine the object's speed at different points in its trajectory, such as just before impact.

Conservation of Mechanical Energy

This principle states that in the absence of non-conservative forces (like friction or air resistance), the total mechanical energy of a system (the sum of kinetic and potential energies) remains constant. In problems involving gravitational fields, this concept allows us to equate the initial energy state with the final energy state, making it possible to calculate changes in velocity as potential energy is converted to kinetic energy during free fall.

Gravitational Potential Energy

Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It is typically expressed as U = -GMm/r, where G is the gravitational constant, M is the mass of the planet, m is the mass of the object, and r is the distance between the centers of the two masses. This concept is essential for determining how much energy can be converted into kinetic energy as an object moves closer to a massive body like Earth.

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