91Ó°ÊÓ

Kinetic energy is a measure of the energy that an object possesses due to its motion. The formula for kinetic energy (KE) is given by the expression: \( KE = \frac{1}{2} mv^2 \). Here, \(m\) represents the mass of the object and \(v\) is its velocity.

In this exercise, the particle's velocity is expressed in terms of the distance \(x\), with the equation \(v = k \sqrt{x}\), where \(k\) is a constant. By substituting the velocity equation into the kinetic energy formula, we derive a new expression for kinetic energy: \( KE = \frac{1}{2} m (k^2 x) \).

This means the kinetic energy of the particle is dependent on both its mass and the square of the distance \(x\) it has traveled, scaled by the constant \(k\). Understanding how kinetic energy changes with distance helps us analyze how energy is transferred during motion.

Variable velocity indicates that the speed of the particle is not constant over time. Instead, it changes as a function of another variable, which in this case, is the distance \(x\) along the path of motion.

For the given problem, the velocity is provided as \(v = k \sqrt{x}\). This tells us that the particle speeds up as it covers more distance. This relationship is non-linear, meaning the increase in velocity is proportional to the square root of the distance covered.

Navigating problems where velocity changes with position requires integrating the velocity with respect to time or another suitable variable to understand the complete motion dynamics. This understanding of variable velocity is crucial in finding how it affects other physical quantities like kinetic energy and work done.

Work done in physics is related to the amount of energy transferred by a force acting upon an object to move it over a distance. According to the work-energy principle, work is equal to the change in kinetic energy of the object.

In this context, the work done on the particle is calculated by evaluating the change in its kinetic energy, represented as \( W = KE_f - KE_i \). Since the particle starts from rest, the initial kinetic energy \(KE_i = 0\), simplifying the work done to \( W = KE \).

Substituting the expression for kinetic energy derived earlier, we find \( W = \frac{1}{8} m k^4 t^2 \). This final result shows how the motion of the particle and its properties like mass and the constant \(k\) define the overall work done.

Differential equations are an essential tool in modeling situations where quantities are changing. In this exercise, we have a differential equation \( \frac{dx}{dt} = k \sqrt{x} \), which derives from the relationship between velocity and distance.When solving this differential equation, we separate the variables to obtain \( \frac{dx}{\sqrt{x}} = k \, dt \). Integrating both sides helps us find expressions for \(x\) and \(t\), relating distance to time for the motion of the particle.

After integration, we assume initial conditions (e.g., at \(t=0\), \(x=0\)) to determine constants, leading to the relationship \(2\sqrt{x} = kt\) or \(x = \frac{k^2 t^2}{4}\).

Understanding differential equations is vital in capturing the dynamics of variable motion and finding how different physical quantities evolve over time.