Suppose a molecular cloud begins with a mass of 1 M⊙ and has a radius of 1 ly. Imagine a dust gr

Suppose a molecular cloud begins with a mass of 1 M⊙ and has...

Suppose a molecular cloud begins with a mass of $1 M_{\odot}$ and has a radius of 1 ly. Imagine a dust grain at the outer edge of the cloud pulled straight inward, accelerating the entire way. When it reaches a radius of 1 million $\mathrm{km}$ from the center of the forming star, how fast will it be moving? (Hint: This problem is much easier to answer through the conservation of energy.)

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Key Concepts

Gravitational Potential Energy

In a gravitational field, potential energy is the energy stored due to the position of an object relative to another mass. It is typically expressed with a negative sign (U = -GMm/r) to indicate that work must be done to separate the two masses, and changes in this potential energy drive the acceleration of objects falling toward a massive body.

Kinetic Energy

Kinetic energy quantifies the energy of motion and is given by the formula (1/2)mv². As an object falls in a gravitational field, the decrease in gravitational potential energy is directly converted into kinetic energy, which allows us to calculate the object's speed based on the energy difference between its starting and ending positions.

Conservation of Energy

This principle states that in an isolated system, the total energy remains constant over time. In the context of gravitational systems, it implies that gravitational potential energy lost by a body as it falls is converted into kinetic energy, allowing us to determine the speed reached by an object under gravity without tracking its detailed motion along the path.

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