91Ó°ÊÓ

When a spring is compressed or stretched, it stores energy known as spring potential energy. This is a crucial concept in physics as it demonstrates how potential energy gets converted into kinetic energy, propelling an object. The potential energy stored in a spring can be calculated using the formula:

• \( U_s = \frac{1}{2} k x^2 \)

Here, \( k \) is the spring constant, which defines the spring's stiffness, and \( x \) is the displacement, which is how much the spring is compressed or stretched.
In our exercise, the spring constant is 1100 N/m, and it’s compressed by 4 m, resulting in a spring potential energy of 8800 J. This amount of energy will initially propel The Great Sandini forward, acting as the primary source of his kinetic energy. Understanding this exchange of energy is essential in physics problem solving.

The law of energy conservation plays a central role in physics, asserting that energy cannot be created or destroyed, only transformed from one form to another. In our particular scenario, the energy begins as spring potential energy and is ultimately converted into kinetic energy, with some losses due to friction and the increase in gravitational potential energy.
This principle can be illustrated using the equation:

• Initial Energy = Final Energy + Energy Losses

In this exercise, the initial spring energy is set to equal the sum of kinetic energy when The Great Sandini exits the barrel, the work done by friction, and the change in gravitational potential energy. By applying this principle, we ensure that all forms of energy throughout the process are accounted for, providing a comprehensive understanding of the system's dynamics.

Friction is a force that opposes motion, and in this context, it's an energy loss that takes away from the potential energy provided by the spring. The work done by friction can be calculated using the formula:

• \( W_f = f \times d \)

where \( f \) is the friction force, and \( d \) is the distance over which this force is applied.
In the case of The Great Sandini, the friction force is 40 N, and it acts over a 4 m distance, resulting in a friction work of 160 J. This energy, subtracted from the initial spring energy, represents the portion of energy lost due to the opposing force of the Teflon-coated barrel.

Gravitational potential energy is the energy an object possesses due to its height above the ground. This energy is calculated through the formula:

• \( \Delta U_g = mgh \)

Here, \( m \) is mass, \( g \) is the acceleration due to gravity (approximately 9.8 m/s²), and \( h \) is the height above the starting point.
For our circus performer, moving up 2.5 meters while weighing 60 kg, the change in gravitational potential energy is 1470 J. This increase in potential energy while ascending reduces the kinetic energy available for maintaining speed, further emphasizing the interplay between different types of energy in closed systems.

Kinetic energy represents the energy of motion. After the conversion of spring potential energy and accounting for energy losses due to friction and gravity, The Great Sandini's speed can be calculated using kinetic energy. The formula is:

• \( K_f = \frac{1}{2} mv^2 \)

where \( m \) is mass and \( v \) is velocity.
In this exercise, once you defeat frictional and gravitational challenges, you find that the remaining kinetic energy is 7170 J. Substituting this value, along with Sandini’s mass of 60 kg, into the kinetic energy formula allows us to solve for \( v \), yielding a velocity of approximately 15.5 m/s. This final step demonstrates how potential energy transformations and energy losses interact to define movement and speed.