91Ó°ÊÓ

The work-energy principle is a fundamental concept in mechanics. It states that the work done on an object is equal to its change in kinetic energy. In mathematical terms, this is expressed as:

• \(W = \Delta KE\)

Where \(W\) is the work done, and \(\Delta KE\) is the change in kinetic energy. This principle is powerful because it links the dynamics of an object to the work performed on it by forces. In the subway train exercise, this principle allows us to relate the work done by the spring to the train's speed change.When the train comes to a stop, all its kinetic energy is converted into work done by the spring. This is why the equation becomes

• \(\frac{1}{2}kx^2 = -\frac{1}{2}mv_i^2\),

where the minus sign indicates a decrease in energy. This principle provides a handy route to connect various factors like force, movement, and energy transformations.

The spring constant, \(k\), is a measure of how stiff a spring is. It's the restoring force per unit of displacement. When dealing with springs, Hooke's Law is often used, expressed as:

• \(F = kx\)

This states that the force \(F\) exerted by a spring is proportional to the displacement from its equilibrium position, \(x\).In our problem, the work done by the spring can be established as:

• \(W = \frac{1}{2}kx^2\)

This formula is crucial when calculating the spring constant. By equating this to the change in kinetic energy, a formula for \(k\) derived:

• \[ k = \frac{-mv_i^2}{x^2} \]

This equation allows us to substitute known values for mass, initial speed, and stopping distance to find \(k\). Remember, the spring constant should always be positive, reflecting the nature of work and energy in physical systems.

Kinetic energy is the energy an object has due to its motion. It is fundamental in mechanics and is given by the formula:

• \(KE = \frac{1}{2}mv^2\)

Where \(m\) is the mass of the object and \(v\) is its velocity. In this formula, the velocity is squared, meaning that any change in velocity will have a significant impact on the kinetic energy.In the context of the subway train problem, the initial kinetic energy is crucial to our calculations. The train's kinetic energy at its initial speed \(v_i\) is:

• \(KE_i = \frac{1}{2}mv_i^2\)

When the train stops, its kinetic energy is entirely transferred to the spring. Recognizing this energy transfer is essential in applying the work-energy principle and accurately assessing the spring's characteristics.This energy shift illustrates how kinetic energy transforms as objects interact in mechanical systems, highlighting the interconnectedness of energy, motion, and forces.