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Escape velocity is the minimum speed a projectile must have to completely break free from a planet's gravitational pull without any further propulsion. On Earth, the formula to calculate escape velocity is: \[ v_e = \sqrt{\frac{2GM}{R_E}} \]Where:

• \( G \) is the gravitational constant.
• \( M \) is the mass of Earth.
• \( R_E \) is Earth's radius.

Imagine throwing a ball as hard as possible. For it to never come back, it must reach or exceed this escape velocity. This formula helps us understand that Earth's gravity tries to pull everything back but with the right speed, we can overcome it. Escape velocity doesn't depend on the direction you throw the projectile; any direction at the right speed will do.
Understanding escape velocity is crucial for space travel and launching satellites.

Conservation of energy is a fundamental principle of physics. It asserts that energy cannot be created or destroyed, only transformed from one form to another. In the context of projectile motion, it plays a vital role in determining the projectile's journey and how far it can travel. In this exercise, the projectile's initial kinetic energy and gravitational potential energy transform throughout its motion but their sum remains constant. This relationship is shown as: \[\text{initial mechanical energy} = \text{final mechanical energy}\]If a projectile starts with enough energy, it can escape Earth's gravity entirely. Conservation of energy tells us that any decrease in kinetic energy (like when moving upward against gravity) results in a corresponding increase in potential energy. Conversely, when the projectile loses height, its kinetic energy increases as potential energy decreases. This seamless transition between energy types is mediated by the conservation law, guiding the projectile's behavior throughout its journey.

Gravitational potential energy (GPE) is the energy held by an object because of its position relative to Earth. The formula for calculating GPE of an object at distance \( r \) from the Earth's center is:\[U = -\frac{GMm}{r}\]

• \( G \) is the gravitational constant.
• \( M \) is the mass of Earth.
• \( m \) is the mass of the object.
• \( r \) is the radial distance from Earth's center.

GPE is negative because work must be done against the gravitational field to move away from Earth. As you increase the distance \( r \) from Earth, the GPE becomes less negative, climbing towards zero. In our problem, this energy provides insight into how much total energy the projectile retains at different distances from Earth. The greater the radial distance reached, the more GPE it possesses, indicating efficient use of its initial kinetic energy to overcome Earth's gravity.

Kinetic energy (KE) refers to the energy a body possesses due to its motion. It is given by the formula:\[KE = \frac{1}{2}mv^2\]Where:

• \( m \) is the mass of the object.
• \( v \) is its velocity.

In the scenario of projectile motion, kinetic energy characterizes how fast the projectile can initially be launched. The stronger the projectile is shot, the higher the initial velocity, and therefore the greater its kinetic energy. In our example, analyzing kinetic energy helps decide how much more speed is required for the projectile to reach or exceed escape velocity.
Half the escape kinetic energy means the projectile won’t escape but can still reach a significant height due to shared work with gravitational potential energy. The conversion between kinetic and potential energy throughout the projectile's motion is key to understanding its trajectory and ultimate radial reach.