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Kinetic energy is a term we often hear in physics, especially when an object is in motion. It's essentially the energy that an object possesses due to its movement. To calculate the kinetic energy (\textbf{KE}) of an object, we use the formula:
\( KE = \frac{1}{2} mv^2 \)
where \( m \) is the mass of the object, and \( v \) is the velocity at which it's moving. In the context of our rocket exercise, calculating kinetic energy involves just plugging the mass and the velocities into this equation. Initially, when the rocket is launched at 10 m/s, its kinetic energy can be computed as being 2,500 Joules. Later, when the velocity increases to 15 m/s, the kinetic energy elevates to 5,625 Joules. Understanding these calculations is crucial, as they play a key role in applying the work-energy theorem to determine the work done by various forces.

When we say 'work done by forces,' we are referring to a specific physical concept. In physics, work (\textbf{W}) is defined when a force (\textbf{F}) causes a displacement (\textbf{d}) and is calculated using the formula:
\( W = Fd \cos(\theta) \)
Here, \( \theta \) is the angle between the force and the displacement. Work has both magnitude and direction, which is why the concept of positive and negative work arises. In our rocket example, the secondary engines exert a force in the same direction as the displacement, thereby doing positive work which adds energy to the rocket. However, when it comes to gravity, it does negative work because gravitational force and the displacement of the rocket are in opposite directions. This distinction is key to solving problems involving multiple forces acting on an object, as we see with the work done to change the rocket's kinetic energy.

Gravitational work may sound a bit daunting, but it's straightforward when broken down. It refers to the work done by the gravitational force (the weight of the object) as it moves through a distance. Since gravity always acts downward, any motion opposing it (like our rocket moving upwards) results in negative work. In our scenario, the gravitational work can be calculated by taking the total work needed to change the kinetic energy and subtracting the work done by the engines. Essentially, we identify the energy added by the engines and then see what’s left over—that’s the work gravity has done, slowing the ascent of the rocket. In the given exercise, the gravitational work comes out to be -1,125 Joules (negative indicating work done against the motion of the rocket), which is crucial for a thorough understanding of energy changes in physics problems involving gravity.