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Conservation of Energy is a principle stating that in isolated systems, energy can neither be created nor destroyed, only transformed from one form to another. Consider a spring-loaded rifle firing a pellet. Initially, energy is stored as potential energy in the compressed spring. As the spring releases, all of this potential energy transforms into kinetic energy, which propels the pellet upwards. Once the pellet breaks free from the spring, kinetic energy turns into gravitational potential energy as the pellet gains height. Here is what happens:

• Initial: All energy is potential, stored in the compressed spring.
• During motion: Potential energy is converted to kinetic energy of the moving pellet.
• At the peak: Kinetic energy is fully transformed to gravitational potential energy.

In perfect conditions (like in our problem without air resistance), the sum of energies remains constant, ensuring the conservation of energy. This allows us to equate potential energy in the spring to the gravitational potential energy at the pellet's maximum height.

Potential Energy is the energy held by an object because of its position or state. When we talk about a spring, potential energy is stored during its compression or stretching. For a compressed spring, the potential energy can be calculated using the formula:\[ PE_{spring} = \frac{1}{2} k x^2 \]where:

• \(k\) is the spring constant, representing spring stiffness.
• \(x\) is the displacement from its natural length in meters.

The formula tells us how much energy is stored based on how compressed the spring is. The stiffer the spring (higher \(k\)), the more energy it can store. It's this energy that launches the pellet upwards. Understanding the potential energy of the spring helps in knowing the power behind the pellet's launch and how it will convert into other energy types.

Kinetic Energy is the energy of motion. When the spring releases its stored potential energy, this energy converts into kinetic energy, giving the pellet speed. The formula for kinetic energy is:\[ KE = \frac{1}{2} mv^2 \]where:

• \(m\) is the mass of the pellet.
• \(v\) is the velocity at which it moves.

As the pellet is shot upwards, kinetic energy is at its peak right after leaving the spring. As it continues to rise, kinetic energy decreases while gravitational potential energy increases. Even without explicit values of velocity, we understand that this transformation ensures that the pellet ascends until it reaches its maximum height. The conservation principle assures that as kinetic energy decreases, the energy isn't lost but merely converted into another form of energy in the system.