Planet Roton, with a mass of 7.0 ×10^24 kg and a radius of 1600 km, gravitationally attracts a m

step 1 For this problem on the topic of gravitation, we are told that a planet has a mass of 7 times 10

Planet Roton, with a mass of 7.0 ×10^24 kg and a radius of...

Planet Roton, with a mass of $7.0 \times 10^{24} \mathrm{kg}$ and a radius of 1600 $\mathrm{km},$ gravitationally attracts a meteorite that is initially at rest relative to the planct, at a distance great cnough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, find the speed of the meteorite when it reaches the planet's surface.

Planet Roton, with a mass of $7.0 \times 10^{24} \mathrm{kg}$ and a radius of 1600 $\mathrm{km},$ gravitationally attracts a meteorite that is initially at rest relative
to the planct, at a distance great cnough to take as infinite. The meteorite falls toward the planet. Assuming the planet is airless, find the speed of the meteorite when it reaches the planet's surface.

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Key Concepts

Conservation of Energy

This concept states that in an isolated system, the total mechanical energy (sum of kinetic and potential energies) remains constant. In gravitational problems, as an object falls toward a massive body, the drop in gravitational potential energy converts directly into kinetic energy, allowing for the prediction of speeds without directly computing the acceleration at each point along the path.

Gravitational Potential Energy

Gravitational potential energy in a two-body system is defined by the formula -GMm/r, where changes in the distance from the mass center change the stored energy. When an object starts from rest at a point where the potential is taken as zero (at infinity), the gravitational pull increases its kinetic energy by the magnitude of the loss in potential energy as it moves closer to the attracting mass.

Escape Velocity

The concept of escape velocity is the speed at which an object must travel in order to break free from the gravitational influence of a planet without further propulsion. For an object falling from infinity, its impact speed at the surface when starting from rest is equivalent to the escape velocity in reverse, which is mathematically obtained by v = sqrt(2GM/r).

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