91Ó°ÊÓ

Energy conservation is a vital principle in physics which states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of our exercise, the initial elastic potential energy stored in the compressed spring is transferred into The Great Sandini’s kinetic energy as he moves through the barrel.
However, some energy is also transferred to overcome friction and to increase his gravitational potential energy as he rises by 2.5 meters.
The conservation equation used is:

• Initial Elastic Potential Energy = Final Kinetic Energy + Work done against friction + Change in Gravitational Potential Energy

Understanding this equation allows us to determine the speed at which The Great Sandini emerges from the cannon by accounting for all forms of energy transformation involved in his motion.

Kinetic energy is the energy of motion. It's defined by the formula:

• \( \text{KE} = \frac{1}{2} mv^2 \)

where \( m \) is the mass and \( v \) is the velocity. In this exercise, once The Great Sandini exits the spring gun, his kinetic energy is the leftover energy after losses to friction and potential energy changes.
To compute his final speed, we rearrange the formula for kinetic energy to solve for velocity, indicating how fast he travels once all energy transfers are accounted for.

Potential energy is stored energy that an object has due to its position or state. The primary forms considered here are elastic potential energy, from the compressed spring, and gravitational potential energy, due to The Great Sandini's elevated position.
Gravitational potential energy is given by:

• \( \Delta \text{GPE} = mgh \)

where \( m \) is the mass, \( g \) is the acceleration due to gravity, and \( h \) is the height difference. As he rises to a height of 2.5 m, some of the initial energy converts into gravitational potential, reducing the kinetic energy he has available.

Elastic potential energy is energy stored in objects that can be stretched or compressed, like springs. The energy stored in the spring is determined by the formula:

• \( ext{EPE} = \frac{1}{2} k x^2 \)

where \( k \) is the spring constant, and \( x \) is the compression distance of the spring.
In the problem, initial compression of the spring captures energy ready to be converted into kinetic motion.
This form of energy is crucial in powering The Great Sandini's launch and must overcome all energy losses to define his speed.

Work against friction is the energy required to overcome resistive forces that oppose motion. In this context, it refers to the force needed to move The Great Sandini against the friction within the cannon barrel coated with Teflon.
This work is calculated by:

• \( W_{friction} = f \cdot d \)

where \( f \) is the frictional force and \( d \) is the distance moved. Understanding how friction reduces the total energy available for The Great Sandini's motion helps us appreciate its role in real-world energy conversions.
In our exercise, work done against friction decreases the kinetic energy that could otherwise be used for velocity, thus impacting his speed.