What is the minimum kinetic energy an impact must give to a...

What is the minimum kinetic energy an impact must give to a 3.0 - -diameter rock for the rock to beable to escape from the Earth? $$\begin{array}{l}{\text { A. } 5.3 \times 10^{3} \mathrm{J} \quad \text { B. } 1.3 \times 10^{6} \mathrm{J}} \\ {\text { C. } 2.7 \times 10^{12} \mathrm{J} \quad \text { D. } 2.1 \times 10^{13} \mathrm{J}}\end{array}$$

What is the minimum kinetic energy an impact must give to a 3.0 - -diameter rock for the rock to beable to escape from the Earth?
$$\begin{array}{l}{\text { A. } 5.3 \times 10^{3} \mathrm{J} \quad \text { B. } 1.3 \times 10^{6} \mathrm{J}} \\ {\text { C. } 2.7 \times 10^{12} \mathrm{J} \quad \text { D. } 2.1 \times 10^{13} \mathrm{J}}\end{array}$$

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Key Concepts

Mass Determination for Spherical Objects

The mass of a spherical object can be calculated by first determining its volume using the sphere volume formula (4/3 ?r³) and then multiplying by its density. This process is vital because the object's mass directly contributes to the kinetic energy needed for it to escape a gravitational field.

Gravitational Potential Energy

Gravitational potential energy is the energy stored in an object due to its position within a gravitational field. When overcoming gravitational forces, such as for escape, an object must have kinetic energy that exceeds this potential energy, making the concept essential to energy balance calculations in mechanics.

Kinetic Energy

Kinetic energy is the energy an object has due to its motion and is given by the formula (1/2)mv², where m is mass and v is velocity. This concept is central when determining the energy required for an object to achieve a certain motion or to overcome opposing forces, such as gravity.

Escape Velocity

Escape velocity is the minimum speed required for an object to break free from a celestial body's gravitational pull without further propulsion. It is derived from equating the kinetic energy of the object to the gravitational potential energy that binds it, and its calculation is critical in understanding the energy thresholds for escape.

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